integral operators lead to fibrations, given by the kernels of the Hessian of a phase function .... In this case the theory of Fourier integral operators replaces.
Russian Math. Surveys, 55 (2000), 93-161.
Singularities of affine fibrations in the regularity theory of Fourier integral operators Michael Ruzhansky In the paper the regularity properties of Fourier integral operators are considered in different function spaces. The most interesting case are Lp spaces, for which a survey of recent results is made. Thus, the sharp orders are known for operators, satisfying the so-called smooth factorization condition. Further in the paper this condition is analyzed in both real and complex settings. In the last case conditions for the continuity of Fourier integral operators are related to the singularities of affine fibrations in (subsets of) Cn , defined by the kernels of Jacobi matrices of holomorphic mappings. Singularities of such fibrations are analyzed in this paper in the general case. In particular, it is shown that for small dimensions n or for small ranks of the Jacobi matrix, all singularities of an affine fibration are removable. Fourier integral operators lead to fibrations, given by the kernels of the Hessian of a phase function of the operator. Based on the analysis of singularities for operators, commuting with translations, in a number of case the factorization condition is shown to be satisfied, which leads to Lp estimates for operators. In the other cases, the failure of the factorization condition is exhibited by a number of examples. Results are applied to derive Lp estimates for solutions of the Cauchy problem for hyperbolic partial differential operators.
Contents 1 Introduction 1.1 Regularity of Fourier integral operators . 1.2 Fibrations with affine fibers . . . . . . . . 1.3 Formulation for the parametric fibrations 1.4 Several applications . . . . . . . . . . . . 2
3
Estimates for Fourier integral 2.1 Fourier integral operators . . 2.2 Function spaces . . . . . . . 2.3 Estimates . . . . . . . . . . . 2.4 Parametric fibrations . . . .
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14 14 17 20 23
Affine fibrations 27 3.1 Methods of complex analytic geometry . . . . . . . . . . . . . . . . . . 27 3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1
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Affine fibrations of gradient type 4.1 Localization . . . . . . . . . . . . . . . . . . . . 4.2 Fibrations of gradient type . . . . . . . . . . . . 4.3 Reconstruction of the phase function . . . . . . 4.4 Existence of singular fibrations of gradient type
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42 42 45 48 52
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Further estimates for Fourier integral operators 54 5.1 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Operators, commuting with translations . . . . . . . . . . . . . . . . . . 55 5.3 Application to Fourier integral operators . . . . . . . . . . . . . . . . . 56
6
Cauchy problem for hyperbolic equations 58 6.1 Estimates for the solutions . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Monge–Amp`ere equation . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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Sharpness of the estimates 63 7.1 Essentially homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 A representation formula for continuous operators of small negative orders 65
1 1.1
Introduction Regularity of Fourier integral operators
The regularity theory of solutions of partial differential equations has been attracting the attention of mathematicians for a long time. One of important examples is the wave equation, whose solution is well known and admits a representation as a sum of two elliptic Fourier integral operators with phase functions solving the eikonal equation of geometric optics. In general, solutions to the Cauchy problem for hyperbolic equations admit similar representations. This and other applications led to the development of the theory of singular integral operators, pseudo-differential operators and Fourier integral operators. In contrast with elliptic equations, singularities of solutions of hyperbolic equations propagate along the singularities of the Schwartz integral kernel of the solution operator. In this case the theory of Fourier integral operators replaces the theory of pseudo-differential operators, which was sufficient for elliptic equations. In this paper we will discuss the regularity theory of non-degenerate Fourier integral operators in Lp and other function spaces. In this paper by the continuity in Lp (or from Lp to Lp ) we will understand the continuity (of a linear operator) from Lpcomp to Lploc . Because Lp results may depend on the geometric structure of the wave front of the operator, we will discuss several related problems from the singularity theory of the wave fronts. Singularities arising in the Lp theory are a particular case of the singularities of affine fibrations. For the latter we will derive estimates on the dimensions of the set of its essential singularities and will analyze its structure. Let T be a Fourier integral operator. This means that locally T is of the form Z Z T u(x) = eiΦ(x,y,θ) a(x, y, θ)u(y)dθdy, (1.1) Rm
RN
2
where a ∈ S ν (Rn × Rm × RN ) is a symbol of order ν. This means that a is a smooth function for which the estimate β ∂ξα a(x, y, θ)| ≤ C(α, β, K)(1 + |θ|)ν−|α| |∂x,y
(1.2)
holds in every compact set K and all multi indices α and β. The phase function Φ is smooth, nondegenerate, and positively homogeneous of degree one in θ. We should note that we will give a more general definition of general symbols in Sρμ and phase functions on arbitrary smooth manifolds in Section 2.1. The Schwartz integral kernels of operators of the form (1.1) are called the Lagrangian distributions (or Fourier integral distributions). The wave front of the Lagrangian distribution of an operator T defines a geometric invariant for the operator T . Indeed, the set W F (T ) = {(x, dx Φ(x, y, θ), y, dy Φ(x, y, θ)) : dθ Φ(x, y, θ) = 0}
(1.3)
in the cotangent bundle T ∗ (Rn ×Rm ) does not depend on the choice of a phase function Φ. If T ∗ (Rn × Rm ) is equipped with its standard symplectic form, then the conic set in (1.3) becomes a Lagrangian submanifold of the cotangent bundle T ∗ (Rn × Rm ). The converse statement is one of the main results of the global theory of Fourier integral operators. It can be formulated more conveniently in the manifold setting. Let X and Y be real smooth manifolds of dimensions n and m, respectively, and in this paper we will assume that n = m. Let σX and σY denote the canonical symplectic forms on T ∗ X and T ∗ Y , respectively. Let C be a conic Lagrangian submanifold of the cotangent bundle T ∗ X\0 × T ∗ Y \0 equipped with the symplectic form σX ⊕ −σY . Then C defines a family of Fourier integral operators T with W F (T ) = C, locally of the form (1.1). The set C is called the canonical relation. If we fix an order ν of a symbol a in (1.1), the family of operators T is denoted by I μ (X, Y ; C), where μ = ν + (N − n)/2. Let us give now several important examples of Fourier integral operators. If we take n = m = N , a = 1, and Φ(x, y, θ) = hx − y, θi, (1.4) then the right hand side of (1.1) is a composition of the Fourier transform and its inverse in Rn , and T is the identity operator in this case. If a phase function Φ of an operator T is given by (1.4) and its symbol a is polynomial in θ, then T defines a partial differential operator with symbol a. If a phase function Φ of T is given by (1.4) and its symbol a is arbitrary, as in (1.2), then T defines a pseudo-differential operator with symbol a. The space of pseudo-differential operators of order μ is denoted by Ψμ . The solution operator to the wave equation has the phase function of the form Φ(x, y, θ) = hx − y, θi + |θ|. In Section 5.2 we will give more examples of convolution operators and in Section 6 the solution operators to the Cauchy problem for hyperbolic equations will be regarded as Fourier integral operators as well. In this paper we will be interested in the continuity properties of the described operators in various function spaces. The best behavior is exhibited by pseudo-differential operators. Thus, pseudo-differential operators P ∈ Ψ0 of zero order are continuous (as linear operators) from Lp to Lp for all 1 < p < ∞. Moreover, a pseudo-differential operator P ∈ Ψμ of order μ ∈ R can be extended to a continuous operator from the Sobolev space Lpk to Lpk−μ for all k ∈ R, k ≥ μ, and 1 < p < ∞. Similar results hold in Lipschitz spaces, operators P ∈ Ψμ are continuous from Lip (γ) to Lip (γ − μ) for all γ > μ. However, the phase function Φ for pseudo-differential operators is of the 3
form (1.4) and its canonical relation C is equal to the conormal bundle to the diagonal in Rn × Rn . For the Fourier integral operators the structure of the canonical relation C is much more complicated and their continuity properties depend on the geometric structure of the corresponding canonical relations. Let T ∈ I 0 (X, Y ; C) be a Fourier integral operator of zero order, associated to the canonical relation C. Let πX×Y , πX , πY be the canonical projections: π
X T ∗ X\0 ←−
π
Y C −→ π y X×Y X ×Y
T ∗ Y \0.
(1.5)
It turns out that the continuity properties of a Fourier integral operator T rely heavily on singularities of the projections πX×Y , πX , πY . Projections πX , πY can be diffeomorphic only simultaneously and in this case for every λ0 = (x0 , ξ0 , y0 , η0 ) ∈ C there exists a symplectomorphism χ (a diffeomorphism preserving the symplectic structure) in a neighborhood of the point (y0 , η0 ) ∈ T ∗ Y \0 such that in a neighborhood of λ0 , the canonical relation C has the form {(x, ξ, y, η) : (x, ξ) = χ(y, η)}.
(1.6)
In this case, C is locally equal to the graph of a canonical symplectic transformation and is called a local canonical graph or just a local graph. It is clear that C being a canonical graph implies that the projections πX , πY are diffeomorphic from C to T ∗ X\0 and T ∗ Y \0, respectively. In particular, this implies n = m. The converse is also true. In fact, assume that, say, πY : C → T ∗ Y \0 is a local diffeomorphism. Then, the canonical relation is locally of the form (1.6) in (y, η) coordinates and the condition of C to be Lagrangian for σX ⊕ −σY implies that σX ⊕ −σY vanishes on C and σY = χ∗ (σX ). The latter means that χ∗ is a symplectomorphism and C is a canonical graph. In this paper we are interested in applications to the hyperbolic partial differential equations, where canonical relations are local canonical graphs. Such operators arise as solution operators of hyperbolic Cauchy problems. In this case, the mapping πX |C ◦πY−1 is equal to χ in (1.6) and defines a local diffeomorphism from T ∗ Y \0 to T ∗ X\0. It follows that dimensions of X and Y coincide. Operators T ∈ I 0 (X, Y ; C) with a local canonical graph C are continuous in L2 ([14], [32], [33]). The proof is based on the fact that the canonical relation of T ◦ T ∗ is the conormal bundle of the diagonal in X × X, and, therefore, it is a pseudo-differential operator of order 0 and hence bounded on L2 . In general, for 1 < p < ∞, p 6= 2, Fourier integral operators of order zero need not be bounded on Lp . From the point of view of the Lp continuity of Fourier integral operators, pseudo-differential operators and operators arising as solution operators to the wave equation are two opposite cases. The phase function of the latter has the form hx − y, ξi + |ξ| in Rn , and Littman ([37]) has shown that the corresponding operators T ∈ I μ (X, Y ; C) are not bounded in Lp when μ > −(n − 1)|1/p − 1/2|. The Lp properties of solutions to hyperbolic Cauchy problems for the equations of the wave type have been studied in many papers ([59], [43], [40], [15]). Lipschitz and Lp estimates for the wave equation on compact manifolds were derived in [19] and some results for hyperbolic equations are in [63]. General results on Lp continuity were obtained in [56]. Let us describe them in more detail. Operators T ∈ I μ (X, Y ; C) are bounded from Lpcomp to Lploc 4
if μ ≤ −(n − 1)|1/p − 1/2| and this order is sharp if T is elliptic and dπX×Y |C has full rank, equal to 2n − 1, anywhere. These conditions hold for operators arising as solutions to strictly hyperbolic Cauchy problems in some cases. There is the same loss of order by (n − 1)|1/p − 1/2| in Sobolev and in Lipschitz spaces (in Lipschitz spaces p = ∞). If we denote αp = (n − 1)|1/p − 1/2|, then operators T ∈ I μ (X, Y ; C) are continuous from Lpα to Lpα−αp −μ and from Lip (α) to Lip (α − α∞ − μ). The proof of the Lp boundedness is based on the complex interpolation method. Having the L2 boundedness of zero order operators, the problem reduces to showing that operators of order −(n − 1)/2 are locally bounded from the Hardy space H 1 to L1 . Hardy spaces will be discussed in Section 2.2 in more detail. In general, the projection πX×Y satisfies inequalities n ≤ rank dπX×Y |C ≤ 2n − 1,
(1.7)
because C is conic. The boundary cases are pseudo-differential operators with no loss of smoothness and solutions to strictly hyperbolic partial differential equations with the loss of (n − 1)/2 derivatives. An important ingredient influencing the best order for Lp continuity is the rank in (1.7). Here it is essential that p 6= 2 because the L2 results do not depend on the rank in (1.7). Now, we will formulate the important result due to [56] for the operators in Iρμ (X, Y ; C), which assures the improved Lp regularity under the following condition. The canonical relation C will be said to satisfy the smooth factorization condition, if there exists k, 0 ≤ k ≤ n − 1, such that πX×Y can be locally factored by fiber–preserving homogeneous maps on C of constant rank n + k. More precisely, this means that for every λ0 = (x0 , ξ0 , y0 , η0 ) ∈ C there is a conic neighborhood Uλ0 of λ0 in C, and a smooth map πλ0 : C ∩ Uλ0 → C, homogeneous of degree 0, such that rank dπλ0 πX×Y |C∩Uλ0
≡ =
n + k, πX×Y ◦ πλ0 .
(1.8)
Under the smooth factorization condition, the operators in I μ (X, Y ; C) are bounded from Lpcomp to Lploc provided 1 < p < ∞ and μ ≤ −k|1/p − 1/2|. We will discuss the sharpness of these orders in Section 7 and will give general results for the class Iρμ in Section 2.3. The smooth factorization condition is not necessary for operators T ∈ I μ with μ ≤ −k|1/p − 1/2| to be continuous in Lp . Several examples show that the continuity is possible when the factorization condition fails ([55]). The relaxation of the smooth factorization condition is an open problem. The best result would be to show that operators T ∈ I μ (X, Y ; C) are bounded from Lpcomp to Lploc provided μ ≤ −k|1/p − 1/2|, 1 < p < ∞, and rank dπX×Y |C ≤ n + k. Note that the proof of this result for k = n − 1 is based on the continuity result from H 1 to L1 for operators of the order −(n − 1)/2. One can assume that these operators are also weakly continuous in L1 . However, this problem is still open. The factorization condition is interesting in its own right and it allows fascinating generalizations, which we will discuss in Sections 3 and 4. The smooth factorization condition is trivially satisfied in two cases: pseudo-differential operators with k = 0 and the maximal rank case with k = n − 1, where πλ0 is the projection along the conical direction. It turns out, that under some natural conditions on the canonical relation C, corresponding to the most important cases, it is extremely difficult to exhibit the failure of the smooth factorization condition. 5
Now, we will describe the geometric meaning of this condition. On a smooth submanifold Σ∞ of the set Σ = πX×Y (C) the canonical relation can be found back as the conormal bundle N ∗ Σ∞ of Σ∞ . The set Σ is the singular support of the Schwartz integral kernel of the operator T . The conormal bundle N ∗ Σ∞ of Σ∞ ⊂ X × Y is defined by N ∗ Σ∞
=
{(x, ξ, y, η) ∈ T ∗ (X × Y ) : (x, y) ∈ Σ∞ , ξ(δx) + η(δy) = 0, ∀(δx, δy) ∈ T(x,y) Σ∞ }.
(1.9)
Thus, the diagram N ∗ Σ∞ y
⊂C
πX×Y
(1.10)
Σ∞
defines a smooth local fibration over Σ∞ with affine fibers. In these terms the factorization condition becomes equivalent to the condition that the smooth fibration of N ∗ Σ∞ allows a smooth extension to C. This smooth extension is defined by the levels of the mapping πλ0 . In Sections 3 and 4 we will analyze this property with different degrees of smoothness of fibrations. However, the theory becomes much more subtle if we assume that the described fibration over Σ∞ is analytic. The analyticity assumption is quite natural and is almost always satisfied. In fact, if a critical point of a phase function has finite order of degeneracy, then the phase function is actually even a polynomial in a suitably chosen coordinate system in a neighborhood of the critical point. On the other hand, there are very few infinitely degenerate critical points, because the coefficients of the Taylor expansion of such phase function have to satisfy infinitely many independent algebraic equations. Another reason is that the study of Fourier integral operators is often reduced to the asymptotics of the oscillatory integrals corresponding to the associated Lagrangian distributions. However, the asymptotic behavior (as R λ → ∞) of the oscillatory integrals eλφ ψ depends essentially on the first terms of Taylor expansion, namely the power of the highest order term of the asymptotic expansion corresponds to the first non-vanishing term of the Taylor expansion ([20], [2]). In particular, the analyticity assumption is satisfied for the propagation operators of hyperbolic partial differential equations with analytic coefficients. The properties of the oscillatory integrals with analytic phase functions were studied in, for example, [2]. The corresponding geometric constructions can be found in [33], [4]. The singularities of wave fronts are analyzed in [1]. The smooth factorization condition is called the holomorphic factorization condition if all mappings in (1.8) are analytic (and hence holomorphic after the continuation to a complex domain). We will show, that under the analyticity assumption the holomorphic factorization condition holds in the most important cases. Some results of this type can be found in [50] and [51]. There are alternative presentations of Lagrangian distributions which make use of complex-valued phase functions. Fourier integral operators associated to such functions have been studied in [38] and global representations of Lagrangian distributions were obtained in [36]. A survey of the main theory is in [4]. The converse sharpness results for the wave-type equations can be found in [40], [43], [56] for the case when rank dπX×Y |C = 2n−1 somewhere. For essentially homogeneous symbols (Sρμ with ρ = 1) we will generalize the sharpness results to arbitrary ranks. We 6
will show that the order −k|1/p − 1/2| is sharp for all elliptic Fourier integral operators provided rank dπX×Y |C ≤ n + k. As a consequence it follows that elliptic operators of small negative orders which are continuous in Lp or from Lp to Lq can be obtained as a composition of pseudo-differential operators with Fourier integral operators induced by a smooth coordinate change. Some results of this type appeared in [52] and we will describe them in Section 7. There, we will also mention the case of an arbitrary ρ where the sharpness of the orders for general elliptic operators under a rank restriction condition for the projection πX×Y |C is not settled in general. In Section 5 we will provide examples of the failure of the factorization condition in general and for the canonical relations corresponding to the translation invariant operators in Rn . Now we will briefly describe the smooth factorization condition in terms of the phase function of a Fourier integral operator. From now on, we will replace the frequency variable θ by ξ in the cases when the dimensions of X, Y and the frequency space Ξ coincide. By the equivalence-of-phase-function theorem (Theorem 2.3 below), we can assume that a phase function Φ of an operator T ∈ I μ (X, Y ; C) is of the form Φ(x, y, ξ) = hx, ξi − φ(y, ξ). Therefore, the corresponding wave front is given by ΛΦ = {(∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ))}
and C = Λ0Φ = {(x, ξ, y, −η) : (x, ξ, y, η) ∈ ΛΦ }. The local graph condition is equivalent to (1.11) det φ00yξ (y, ξ) 6= 0 on the support of the symbol of the operator T . The mapping
γ(y, ξ) = Y × Ξ 3 (y, ξ) 7→ (∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ)) ∈ T ∗ X × T ∗ Y defines a diffeomorphism from Y × Ξ to C. The level sets of the mapping πX×Y : C → X × Y correspond to the kernels of the linear mapping dπX×Y |C , or to the kernels of the mapping dπX×Y ◦ dγ. It is straightforward that ker dπX×Y ◦ dγ(y, ξ) = (0, ker
∂2φ (y, ξ)). ∂ξ 2
Therefore, fibration (1.11) reduces to the fibration defined by the kernels ker φ00ξξ (y, ξ), or by the level sets of the mapping (y, ξ) 7→ ∇ξ φ(y, ξ) on the set where the rank of φ00ξξ is maximal. A simple example of the failure of the factorization condition is already possible in R3 . The fibers of the function φ(y, ξ) = hy, ξi +
1 (y1 ξ1 + y2 ξ2 )2 , ξ3
(1.12)
(i.e. the level sets of ∇φ(y, ξ) with respect to ξ) are straight lines with the slope equal to y2 /y1 . It is clear that the corresponding fibration is not continuous at zero. In the case when a phase function is real analytic we will concentrate on its complex extension and on the holomorphic mapping (y, ξ) 7→ ∇ξ φ(y, ξ). Let us discuss the corresponding singularity problem in terms of affine fibrations in more detail. 7
1.2
Fibrations with affine fibers
First, we formulate the problem in the invariant case corresponding to the Fourier integral operators commuting with translations. In this case we have φ(y, ξ) = hy, ξi − H(ξ). Functions H : V → R on an open subset V of Rn have the following property. The maximal rank k of the Hessian D2 H(ξ) is strictly less than n and points ξ where rank D2 H(ξ) = k is maximal form an open set U . One of the interesting properties of the gradient Γ : ξ 7→ ∇H(ξ) is that for every point ξ ∈ U the level set Γ−1 (Γ(ξ)) locally in a neighborhood of ξ coincides with the affine space ξ + ker D2 H(ξ). If H is real analytic, then the holomorphic extension Γ of the gradient H has the same property in some open neighborhood of the open set V in Cn . This property motivates the study of holomorphic mappings Γ with properties (A1)–(A3) formulated below. Let Γ be a holomorphic mapping from a connected open subset Ω of Cm × Cn to p C . Let DΓ(ξ) ∈ Cp×n denote the Jacobian of Γ. The rows of DΓ have the form Dξ Γi (ξ) = (∇ξ Γi )T for 1 ≤ i ≤ p. Let k be such that (A1) rank DΓ(ξ) ≤ k for all ξ ∈ Ω. (A2) ∃ξ ∈ Ω with rank DΓ(ξ) = k. Because we will be interested in the level sets of the mapping Γ, we may assume k ≤ n − 1. The set Ω can be stratified in the union of disjoint sets Ω (i) consisting of points ξ ∈ Ω with rank DΓ(ξ) = i, i = 0, . . . , k. The set Ω\Ω(k) where the rank of DΓ(ξ) < k is an analytic subset of Ω. The mapping κ : ξ 7→ ker DΓ(ξ)
(1.13)
is holomorphic from the open dense in Ω set Ω(k) to the Grassmannian Gn−k (Cn ). Let Ωsing denote the set of essential singularities of the mapping κ, i.e. the set of points ξ ∈ Ω\Ω(k) such that κ does not allow a holomorphic extension over ξ. The mapping κ is regular (holomorphic) in ξ ∈ Ω(k) and such points are called regular. The set Ω\Ω(k) consists of essential singularities (points of Ωsing ) and removable singularities in Ω\(Ω(k) ∪ Ωsing ). An additional assumption on the mapping Γ will be that the affine spaces ξ + κ(ξ) define a fibration in Ω(k) . Let us explain this in more detail. Let ξ ∈ Ω(k) . Then, by the implicit function theorem, the level sets Γ−1 (Γ(ξ)) are smooth analytic manifolds of dimension n − k. The tangent plane to the level set Γ−1 (Γ(ξ)) in the point ξ is given by ξ + κ(ξ). The condition that κ defines a fibration means that each level set coincides with its tangent space at ξ, for all ξ ∈ Ω(k) .
(A3) For every ξ ∈ Ω(k) there exists a neighborhood U of ξ such that the affine space ξ + κ(ξ) coincides with the level set Γ−1 (Γ(ξ)) in U .
Thus, Γ is constant on ξ + κ(ξ) which is equivalent to saying that κ(ξ + ζ) = κ(ξ) for all ξ ∈ Ω(k) and ζ ∈ κ(ξ) with ξ + ζ ∈ Ω(k) (cf. Proposition 3.18 below). Because we are interested in local properties of the mapping κ, we will always assume that Ω is convex. In this case the set (ξ + κ(ξ)) ∩ Ω is connected and Γ is constant on (ξ + κ(ξ)) ∩ Ω. It also means that κ(ξ) = κ(η) for all η ∈ (ξ + κ(ξ)) ∩ Ω(k) . Therefore, condition (A3) implies its global form: (A30 ) For every ξ ∈ Ω(k) , the affine space (ξ + κ(ξ)) coincides with the level set Γ−1 (Γ(ξ)) in the whole of Ω. 8
The mapping κ is holomorphically extendible to the open set Ω\Ωsing containing Ω(k) . This extension will be also denoted by κ. Thus, for every ξ ∈ Ω\Ωsing , Γ is also constant on (ξ + κ(ξ)) ∩ Ω. And, if ξ, η ∈ Ω\Ωsing , then (ξ + κ(ξ)) ∩ (η + κ(η)) ∩ (Ω\Ωsing ) is empty, or, if not, κ(ξ) = κ(η). This enables us to introduce an equivalence relation in Ω\Ωsing . Let us say that the points ξ and η are equivalent if η ∈ ξ + κ(ξ). The element κ(ξ) = κ(η) defines their common equivalence class. The factor space (Ω\Ωsing )/ ∼ is a smooth analytic manifold of dimension k and the projection ξ 7→ (ξ + κ(ξ)) ∩ (Ω\Ωsing ) is an analytic submersion. In this sense spaces (ξ + κ(ξ)) ∩ (Ω\Ωsing ) with ξ ∈ Ω\Ωsing define a smooth fibration in Ω\Ωsing . The mapping Γ factorizes through this fibration (see the beginning of section 3.3). In view of this, the mapping κ itself will be sometimes called the fibration. The simplest singular fibration can be defined for Ω = Cn by taking for one dimensional fibers open straight rays emanating from the origin in radial directions. It is clear that this fibration is regular (analytic) in all points except the origin. The set of its essential singularities is Ωsing = {0}. However, we will show in Section 3 that in this case there exists no holomorphic mapping Γ for which the described rays are given by the kernels of the Jacobian DΓ (even locally). Note that conditions (A1)–(A3) can be formulated in greater generality. In the sequel, we will show that the mapping κ is meromorphic and this allows to put the problem in a general framework of meromorphic mappings. For more generality, one can work with morphisms κ from an analytic set X to a (compact) analytic set Y , such that κ is holomorphic on the complement in X to an analytic set of strictly lower dimension. However, in this paper we are primarily interested in the properties of the mapping κ for a given fixed mapping Γ. This is the reason for us to avoid the general category language and to restrict to the local properties of κ.
1.3
Formulation for the parametric fibrations
Let us briefly consider a more general problem allowing y dependence. Let Γ be a holomorphic mapping from an open connected set Ω ⊂ Cm × Cn to Cp . Let k < n be the maximal rank of the Jacobian Dξ Γ(y, ξ) in Ω, (R1–R2) max(y,ξ)∈Ω rank Dξ Γ(y, ξ) = k. Let Ω(k) be the set of points (y, ξ) where the maximal rank k is attained. It is open and dense in Ω. The mapping κ : (y, ξ) 7→ ker Dξ Γ(y, ξ) is holomorphic from Ω(k) to the Grassmannian Gn−k (Cn ) of (n − k)-dimensional linear subspaces of Cn . Let Ωsing denote the set of the essential singularities of κ, i.e. the set of points (y, ξ) ∈ Ω such that the mapping κ does not allow a holomorphic extension to any neighborhood of (y, ξ) in Ω. It is clear that the sets Ω(k) and Ωsing are disjoint. The following condition guarantee the linearity of the level sets of Γ with respect to ξ: (R3) For every (y, ξ) ∈ Ω(k) the affine space (y, ξ) + (0, κ(y, ξ)) locally coincides with the level set Γ−1 (Γ(y, ξ)) through the point (y, ξ). Similar to conditions (A1)–(A3), (R3) means that κ defines a local holomorphic fibration in Ω\Ωsing . Example (1.12) with Γ = ∇φ shows that in general fibrations can 9
have essential singularities. If we fix a value of y or if we take Γ independent of y, then conditions (R1)–(R3) are equivalent to conditions (A1)–(A3) above. It turns out that simple examples as in (1.12) are impossible in the problem (A1)–(A3) and the analysis becomes more interesting. For example, one of the necessary conditions for an analytic set Ωsing to be the set of the essential singularities of a mapping κ associated to a holomorphic mapping Γ is the following dimension estimate: max{k − 1, n − k + 1} ≤ dimξ Ωsing ≤ n − 2. In particular, Ωsing can not contain isolated points. There is a number of interesting problems related to fibrations arising in this way. Thus, for a given fibration κ0 in an open dense set Ω0 in Ω, we would like to determine whether there exists a holomorphic mapping Γ satisfying conditions (A1)–(A3), such that the fibration defined by κ coincides with the fibration defined by κ0 in Ω0 ∩ Ω(k) . The construction of a phase function φ for a given Γ = ∇ξ φ leads to further complications. In Section 4 we will derive necessary and sufficient conditions in terms of a system of partial differential equations with coefficients corresponding to a given fibrations. However, the regularity (analyticity) of the fibration does not immediately imply that solutions of the constructed system of differential equations are sufficiently regular. Moreover, this system depends on the choice of a local coordinate system in the Grassmannian. It would be interesting to obtain an invariant description of the results of section 4 as well as their generalization to spaces of higher dimensions. The results of section 3 (for example, Theorems 3.9 – 3.12) describe possible dimensions of the set Ωsing under conditions (A1)–(A3). They also give some understanding of its structure. It would be interesting to investigate its further properties, especially in the case of gradient fibrations. For example, in Section 4.4, we will give examples of fibrations of gradient type for which Ωsing is not empty. However, in all our examples, the set Ωsing is affine and its dimension is equal to n − 2. It is not clear whether the condition dim Ωsing = n − 2 is necessary for the fibrations of gradient type. In view of the similarity of two problems described above we will use the same notations in their analysis. In order to eliminate any confusion, we will always consider problem (A1)–(A3) unless we explicitly state otherwise.
1.4
Several applications
One of the main applications of the theory of Fourier integral operators is the theory of hyperbolic partial differential equations. Let P (t, x, ∂t , ∂x ) = ∂tm +
m X
Pj (t, x, ∂x )∂tm−j
j=1
be a strictly hyperbolic operator of order m in a set of points (t, x) ∈ R1+n . The hyperbolicity means that the principal symbol p(t, x, τ, ξ) of the operator P (p is polynomial in τ or degree m and is often denoted by σP ) has m distinct real roots τj in τ . In this case, the Cauchy problem P u(t, x) = 0, t 6= 0, (1.14) ∂tj u|t=0 = fj (x), 0 ≤ j ≤ m − 1. 10
is well posed, and for small t its solution can be written as a sum of Fourier integral operators Ttjl : m m−1 X X jl Tt fl , u(t, ∙) = j=1 l=0
Ttjl
where each operator depends smoothly on t, is of order −l and depends on the root τj (t, x, ξ). This construction will be described in more detail in section 6. Note that operators Ttjl satisfy the local graph condition and it follows from some of the results below (for example, Theorem 2.9 and [56]) that for small t and 1 < p < ∞ for the solution u of the Cauchy problem holds u(t, ∙) ∈ Lploc provided that the Cauchy data fl are in Sobolev spaces fl ∈ (Lpαp −l )comp , where αp = (n − 1)|1/p − 1/2|. Thus, there is a loss of smoothness by αp derivatives in Lp spaces. It follows from the stationary phase method that the loss of αp derivatives is sharp in a number of cases. For example, it is sharp for equations of the wave type with variable coefficients (when m = 2). It is also sharp if one imposes an additional condition that for almost all t at least one of the roots τj (t, x, ξ) is elliptic in ξ, i.e. τj (t, x, ξ) 6= 0 for ξ 6= 0. However, in a number of cases the order αp can be improved. Let k denote the minimal integer for which the ranks of standard projections from the canonical relations of operators Ttjl to Rn × Rn do not exceed n + k for all j and l. These projections are discussed in more detail in Section 1.1. In the present case, projection πX×Y in (1.7) satisfies the inequality rank dπX×Y |C jl ≤ n + k, where X and Y are (open subsets of) Rn and Ctjl are the t
canonical relations of operators Ttjl . It terms of phase functions it means that if Φj is the solution of the eikonal equation (∂/∂t)Φj (t, x, ξ) = τj (t, x, ∇x Φj ) with initial 2 condition Φj (0, x, ξ) = hx, ξi, then rank ∂ξξ Φj (t, x, ξ) ≤ k for all j, x, ξ. In this case one can show that the order αp = k|1/p − 1/2| would be sharp, i.e. the condition that for some α holds u(t, ∙) ∈ Lploc for all fl ∈ (Lpα−l )comp implies α ≥ αp = k|1/p − 1/2|. The converse is known in a number of special cases only. For operators with constant coefficients (in particular) the converse result will be proved in section 6 for k ≤ 2. The complete picture of the Lp properties of such operators becomes clear in the 5dimensional space. We will describe it in section 6. For 3 ≤ k ≤ n − 2 the sharp Lp are not proved in general. In the case k = 0 there is no loss of smoothness for the solutions (in this case the regularity properties of solutions are the same as the regularity properties of solutions to the elliptic Cauchy problems, αp = 0 and, therefore, k = 0), the principal symbol of P has a special form. Its roots τj are linear in ξ (cf. Theorem 6.4). Another important applications of Fourier integral operators are convolution operators and Radon transforms. Let X and Y be smooth manifolds and let S be a smooth submanifold of X × Y . For x ∈ X, let Sx be the set of points y ∈ Y such that (x, y) ∈ S. Let σ be a smooth measure on S. It naturally induces the measure σx on Sx . The Radon transform corresponding to S and σ is now defined by Z Rf (x) = f dσx . Sx
It defines an integral operator R : C0∞ (Y ) → D0 (X). The Schwartz integral kernel of the operator R is a Dirac measure supported in S and equal to σ there. Because we are interested in local properties, we can assume that the set S is compact. Let 11
dim S = dim X + dim Y − d. Locally S is given by d equations h1 (x, y) = . . . = hd (x, y) = 0, where functions hj are smooth and their gradients are linearly independent on S. In this way, the operator R can be regarded as a Fourier integral operator with the phase function d X θj hj (x, y). j=1
The canonical relation of R is then equal to the conormal bundle N ∗ S. The local graph condition means that dim X = dim Y and that projections πX , πY from N ∗ S to T ∗ X and T ∗ Y are locally diffeomorphic. In this case the Lp estimates of the subsequent sections can be applied. The smooth factorization condition is satisfied for N ∗ S because S is a smooth manifold. Therefore, the Lp estimates follow from the general theory of Fourier integral operators. The results depend on the order of the operator R as a Fourier integral operator. Let us assume for simplicity that Rf = f ∗ σ, where the measure σ is supported on some (smooth) submanifold Σ of Rn . The connection between R and Fourier integral operators will be described in detail in Section 5.2. Now, we restrict ourselves to some remarks only. The order of R as a Fourier integral operator is related to the order of decrease of the Fourier transform σ b at infinity. This order depends on the curvature of Σ. Let Σ be a hypersurface (dim Σ = n−1) with nonvanishing Gaussian curvature at all points and let dσ = ψdμ, where dμ is induced on S by the Lebesgue measure and ψ ∈ C0∞ (Rn ). One can show that the Fourier transform of the measure σ satisfies the estimate |b σ (ξ)| ≤ C|ξ|−(n−1)/2 . It follows that R is a Fourier integral operator of order −(n−1)/2. There are generalizations of this estimate to (n−d)–dimensional manifolds Σ of finite type. Roughly speaking, Σ has type m if the order of intersection of Σ with affine hyperplanes does not exceed m. In this case holds |b σ (ξ)| ≤ C|ξ|−1/m . If the manifold Σ is analytic, then, in turn, this estimate implies that Σ has type m. Indeed, otherwise σ b would be constant in directions orthogonal to the tangent space to Σ at the points of infinite order of tangency. The detailed discussion as well as other forms of the Radon transform can be found, for example, in [60], [27], [44], [45], [47]. For the failure of the smooth factorization condition it is necessary (but not sufficient, see Remark 2.23) that S is not smooth. In this case there are additional problems in determining the order of R as a Fourier integral operator since the estimates of σ b at infinity are more complicated. The local graph condition holds in a number of important cases. For example, Radon transforms along hypersurfaces in Rn satisfy this condition. It also holds for the convolution operators with measures supported on hypersurfaces in Rn with nonvanishing Gaussian curvature. As an example, one has solution operators to the Cauchy problem for the wave equation (see above), where for a fixed t > 0, the manifold Sx consists of points y ∈ Rn with |x − y| = t. Let us remark, finally, that in some applications the canonical relation fails to be a local graph. For instance, the local graph condition fails for the Radon transforms with d > n/2 (cf., for example, [44]). For submanifolds of codimension larger than 1 in a general position, the canonical relation C is a Whitney fold ([61], [46], [45]). Projections πX and πY are also singular in applications to the diffraction theory ([64], [65]), in the theory of X-ray transforms ([26]), and deformation theory ([28], [26]). Singular Radon
12
transforms and Fourier integral operators with densities with Calder´ on–Zygmund type of singularities led to the microlocal analysis of the boundedness for the degenerate type of Fourier integral operators (see [29], [27], [45]) and references therein). In many degenerate cases one looses smoothness of functions even in the L2 –case, with the loss dependent on the order of degeneracy, which can be expressed in terms of the stratification of Lagrangian ([46], [47]). A similar loss occurs in Lp norms for the Radon transforms with folding canonical relation and for other types of singularities. However, such degenerate cases fall beyond the scope of the present paper. The bibliography of the paper was selected with the purpose of clarifying some of the aspects of the regularity theory of Fourier integral operators and related geometric problems and it does not pretend to be complete. Many results of this paper were obtained by the author during his work on the doctoral dissertation at Utrecht University. I would like to use the opportunity to thank professor Hans Duistermaat for numerous discussions and his remarks.
2 2.1
Estimates for Fourier integral operators Fourier integral operators
In the sequel, the local properties will be of primary concern. However, global constructions often help to give a better insight in problems at hand. The intrinsic global characterization of Fourier integral operators was systematically developed in [32], [20], [33]. Excellent expositions of the theory can be found in [4], [5]. Many important to us notions are described in [9]. The global theory is based on constructions of the symplectic geometry. Let us recall main definitions. Let M be a smooth real manifold. A form ω is called symplectic on M if it is a 2-form on M such that dω = 0 and such that for each m ∈ M the bilinear form ωm is antisymmetric and nondegenerate on Tm M . The pairs (Tm M, ωm ) and (M, ω) are called the symplectic vector space and the symplectic manifold, respectively. Let X be a smooth real manifold of dimension n. The canonical symplectic form σ on the cotangent bundle T ∗ X of X can be introduced as follows. Let π : T ∗ X 3 (x, ξ) 7→ x ∈ X be the canonical projection. Then for (x, ξ) ∈ T ∗ X the mappings Dπ(x,ξ) : T(x,ξ) (T ∗ X) → Tx X and ξ : Tx X → R are linear. Their composition α(x,ξ) = ξ ◦ Dπ(x,ξ)
(2.1)
∗ is a 1-form on T ∗ X. Its derivative σ = dα is called the canonical 2-form Pn on T X and it follows that σ is symplectic. This form corresponds to the form j=1 dpj ∧ dqj in mechanics, with possible change of sign. It can be shown that any symplectic form takes the latter form in symplectic coordinates. The same objects can be introduced on complex analytic manifolds. A submanifold Λ of T ∗ X is called Lagrangian if (T(x,ξ) Λ)σ = T(x,ξ) Λ, where
(T(x,ξ) Λ)σ = {p ∈ T(x,ξ) (T ∗ X) : σ(p, q) = 0 ∀q ∈ T(x,ξ) Λ}.
In particular, this implies dim Λ = n. A submanifold Λ of T ∗ X\0 = {(x, ξ) ∈ T ∗ X : ξ 6= 0} is called conic if (x, ξ) ∈ Λ implies (x, τ ξ) ∈ Λ for all τ > 0. Let Σ ⊂ X be a smooth submanifold of X of dimension k. Its conormal bundle in T ∗ X is defined by N ∗ Σ = {(x, ξ) ∈ T ∗ X : x ∈ Σ, ξ(δx) = 0, ∀δx ∈ Tx Σ}. 13
The proofs of the subsequent statements can be found in [4], [5], [33]. We mainly follow [22]. Proposition 2.1 (1) Let Λ ⊂ T ∗ X\0 be a closed submanifold of dimension n. Then Λ is a conic Lagrangian manifold if and only if the form α in (2.1) vanishes on Λ. (2) Let Σ ⊂ X be a submanifold of dimension k. Then its conormal bundle N ∗ Σ is a conic Lagrangian manifold. (3) Let Λ ⊂ T ∗ X\0 be a conic Lagrangian manifold and let Dπ(x,ξ) : T(x,ξ) Λ → Tx X have constant rank equal to k for all (x, ξ) ∈ Λ. Then each (x, ξ) ∈ Λ has a conic neighborhood Γ such that (a) Σ = π(Λ ∩ Γ) is a smooth submanifold of X of dimension k,
(b) Λ ∩ Γ is an open subset of N ∗ Σ.
In the sequel we will mainly deal with conic Lagrangian manifolds and we will need their local representations. For this purpose, we consider a local trivialization X × (Rn \0) of T ∗ X\0, where we can assume X to be an open set of dimension n. However, in the sequel we will also need a slight generalization of it, so that we allow the dimensions of X and the fibers differ. Thus, let Γ be a cone in X × (RN \0). A smooth function φ : X × (RN \0) → R is called a phase function, if it is homogeneous of degree one in θ and has no critical points as a function of (x, θ): φ(x, τ θ) = τ φ(x, θ) for τ > 0 and d(x,θ) φ(x, θ) 6= 0 for all (x, θ) ∈ X × (RN \0). A phase function is called nondegenerate in Γ if (x, θ) ∈ Γ, dθ φ(x, θ) = 0 imply that d(x,θ) ∂φ(x,θ) are linearly ∂θj independent for j = 1, . . . , N . Proposition 2.2 (1) Let Γ be a cone in X × (RN \0) and let φ be a nondegenerate e ⊃ Γ such that the set phase function in Γ. Then there exists an open cone Γ e : dθ φ(x, θ) = 0} Cφ = {(x, θ) ∈ Γ
is a smooth conic submanifold of X × (RN \0) of dimension n. The mapping T φ : Cφ 3 (x, θ) 7→ (x, dx φ(x, θ)) ∈ T ∗ X\0 is an immersion, commuting with the multiplication with positive real numbers in the fibers. Let us denote Λφ = T φ (Cφ ). (2) Let Λ be a submanifold of T ∗ X\0 of dimension n. Then Λ is a conic Lagrangian manifold if and only if every (x, ξ) ∈ Λ has a conic neighborhood Γ such that Λ ∩ Γ = Λφ for some nondegenerate phase function φ. Naturally, the cone condition for Λ corresponds to the homogeneity of φ. Let Λ be a closed conic Lagrangian submanifold of T ∗ X\0. A distribution u is called the Lagrangian distribution of order m associated to Λ, u ∈ I m (X, Λ), if n Y
j=1
Pj u ∈
∞
loc H−m−n/4 (X),
14
(2.2)
whenever Pj ∈ Ψ1 (X) are properly supported pseudo-differential operators whose prinloc cipal symbols pj (x, ξ) vanish on Λ and ∞ H−m−n/4 is the localization of the usual 0 n Besov space. First, a distribution u ∈ S (R ) belongs to the Besov space ∞ Hσ (Rn ), if u ˆ ∈ L2loc (Rn ) and ||u|| ∞ Hσ (Rn ) =
Z
2
|ξ|≤1
|ˆ u(ξ)| dξ
!1/2
+ sup j≥0
Z
σj
2j ≤|ξ|≤2j+1
2
|2 u ˆ(ξ)| dξ
!1/2
< ∞.
For a smooth manifold X of dimension n the space ∞ Hσloc (X) is defined to be the space of all u ∈ D0 (X) such that (ψu) ◦ κ −1 is in ∞ Hσ (Rn ) whenever Ω ⊂ X is a coordinate patch with coordinates κ and ψ ∈ C0∞ (Ω). More details can be found in [33, 25.1] and [57, 6.1]. Let now X, Y be open in Rn and let Φ be a nondegenerate phase function. Let a ∈ Sρm (X × Y × RN ) be a symbol of type ρ and order m, which means that 0 ≤ ρ ≤ 1, a ∈ C ∞ (X × Y × RN ), and for every compact subset K of X × Y and any multi-indices α, β holds β ∂ξα a(x, θ)| ≤ C(α, β, K)(1 + |θ|)m−ρ|α|+(1−ρ)|β| |∂x,y for all (x, y) ∈ K and θ ∈ RN \0. Operators of the form Z Z eiΦ(x,y,θ) a(x, y, θ)u(y)dθdy. T u(x) = Y
RN
(2.3)
are called Fourier integral operators. Expression (2.3) can be understood in the classical sense if m + N < 0 and u ∈ C0∞ (X), when the integral is absolutely convergent. The integral kernel of (2.3) is equal to Z eiΦ(x,y,θ) a(x, y, θ)dθ. (2.4) K(x, y) = RN
According to Proposition 2.2, the set Λφ = {(x, y, dx φ(x, y, θ), dy φ(x, y, θ)) : dθ φ(x, y, θ) = 0} is a conic Lagrangian submanifold of T ∗ (X × Y )\0 of dimension 2n. It follows that for ρ = 1, the kernel (2.4) is a Lagrangian distribution in X × Y associated to ΛΦ of order μ = m − n/2 + N/2, K ∈ I μ (X × Y, ΛΦ ). Conversely, any Lagrangian distribution K ∈ I μ (X × Y, Λ) can be microlocally written in the form (2.4) modulo C ∞ , with ρ = 1 (cf. [57, teorema 6.1.4]). The kernel (2.4) is also called the Fourier integral distribution. In general, Iρμ (X, Λ) will denote the space of Fourier integral distributions consisting of Fourier integral distributions Z u(x) = eiφ(x,θ) a(x, θ)dθ, (2.5) RN
with Λ locally equal to Λφ as in Proposition 2.2, a ∈ Sρm (X × RN ), ρ > 1/2, and μ = m − n/4 + N/2. The behavior of the integral (2.3) can be independent of some of the variables θ and the order of T is taken to be μ = m + (N − n)/2, which is the order of its Lagrangian distribution (in this case we have X ×Y instead of X). The following theorem describes 15
the family of phase functions corresponding to the same Fourier integral distribution (2.4) (Theorem 2.3.4 in [22]). We formulate it for the general case of Iρμ (X, Λ) as in (2.5), rather than for a particular case Iρμ (X × Y, Λ). e θ) e are nondegenerate phase functions at Theorem 2.3 Suppose φ(x, θ) and φ(x, e N e (x0 , θ0 ) ∈ X × (R \0) and at (x0 , θe0 ) ∈ X × (RN \0), respectively. Let Γ and Γ e be open conic neighborhoods of (x0 , θ0 ) and (x0 , θ0 ) such that Tφ : Cφ → Γφ and Tφe : Cφe → Γφe are injective, respectively. If Λφ = Λφe, then any Fourier integral distribution, defined by the phase function φ and an amplitude a ∈ Sρm (X × RN ), ρ > 1/2, with ess supp a contained in a sufficiently small conic neighborhood of (x0 , θ0 ), is equal to a Fourier integral distribution defined by the phase function φe and an amplitude e) m+ 12 (N −N
e a ∈ Sρ
e
(X × RN ).
In particular, the phase function Φ of the operator T in (2.3) can be always written in the form Φ(x, y, ξ) = hx, ξi − ψ(y, ξ),
with some function ψ and ξ ∈ Rn . This fact will be often used in the sequel. The function ψ (as well as Φ) is also called the generating function for ΛΦ . Thus, the notion of Fourier integral operator becomes independent of a particular choice of a phase function associated to the Lagrangian manifold Λ. The set C = Λ0 = {((x, ξ), (y, η)) ∈ T ∗ X × T ∗ Y : (x, y, ξ, −η) ∈ Λ}
is a conic Lagrangian manifold in T ∗ X\0 × T ∗ Y \0 with respect to the symplectic structure σX ⊕ −σY and it is called a homogeneous canonical relation from T ∗ Y to T ∗ X. The space of integral operators with distributional kernels in Iρμ (X × Y, Λ) will be denoted by Iρμ (X, Y ; C) and it is the space of Fourier integral operators associated to the canonical relation C = Λ0 .
2.2
Function spaces
In this section we will briefly discuss function spaces, which will be used throughout this paper. Let X be a smooth manifold with a measure λ. For 1 ≤ p < ∞ by Lp (X) we will denote the usual space (of the equivalence classes) of functions f on X with R finite norm ||f ||p = ( X |f |p dλ)1/p . For 0 < p < 1 this expression fails to be a norm and the substitute for Lp (X) in the analysis of singular integrals are Hardy spaces H p (X). The general theory of complex and real variable versions of Hardy spaces can be found in [59], [62], [25], [60], where one can also find proofs of subsequent statements of this section. Since our interest are the local properties, we restrict to the real Euclidean case of H p (Rn ). Let S be the Schwartz space of smooth rapidly decreasing functions, equipped with a countable family of seminorms ||φ||α,β = sup |xα ∂xβ φ(x)|. x∈Rn
Let Φ ∈ S and for t > 0 define Φt (x) = t−n Φ(x/t). Then for a distribution f the convolutions f ∗ Φt are smooth and one defines the maximal operator MΦ f (x) = sup |(f ∗ Φt )(x)|. t>0
16
Let F be a finite collection of seminorms on S and one defines SF = {Φ ∈ S : ||Φ||α,β ≤ 1 for all || ∙ ||α,β ∈ F}. The maximal operator associated to the family SF , determining the approximation of the identity, is now defined by MF f (x) = sup MΦ f (x). Φ∈SF
For the definition of the Hardy space H p (Rn ), we take the space of functions, satisfying one of the following equivalent conditions. Proposition 2.4 Let f be a distribution and let 0 < p ≤ ∞. Then the following conditions are equivalent: R (1) There is a function Φ ∈ S with Φdx 6= 0 so that MΦ f ∈ Lp (Rn ). (2) There is a collection F so that MF f ∈ Lp (Rn ).
The expression ||f ||H p = ||MΦ f ||Lp can be taken to be the norm of H p (Rn ). Note, that it is equivalent to ||MF ||Lp and it is actually a norm only if p ≥ 1. For 0 < p < 1, the topology of H p can be defined by the metric d(f, g) = ||f − g||pH p . In the case 1 < p ≤ ∞, a simple argument shows that H p (Rn ) coincide with the Lebesgue spaces Lp (Rn ). However, already for p = 1 one only has HR1 (Rn ) ⊂ L1 (Rn ). On the other hand, if f ∈ L1comp (Rn ) satisfies the moment condition f dx = 0 (which is in fact necessary for f to belong to H 1 (Rn )), then f ∈ Lq (Rn ) for any q > 1 implies f ∈ H 1 (Rn ). One often makes use of an atomic decomposition of Hardy spaces, similar to the classical Calder´ on-Zygmund decomposition. For 0 < p ≤ 1, an H p atom is a function a such that (1) a is supported in a ball B, (2) |a| ≤ |B|−1/p almost everywhere, R (3) xα a(x)dx = 0 for all α with |α| ≤ n(p−1 − 1).
An H p atom belongs to H p (Rn ) with uniform bound and to Lp (Rn ) with R |a(x)|p dx ≤ 1, which follows from (1) and (2). Proposition 2.5 Let 0 < p ≤ 1.
(1) Let ak be a collection of H p atoms and let λk ∈ C satisfy the series X f= λ k ak
P
k
converges distributionally, its sum f belongs to H p (Rn ), and ||f ||H p ≤ c
17
X k
|λk |
p
!1/p
.
k
|λk |p < ∞. Then (2.6)
(2) Let f ∈ H p (Rn ). Then f can be written as a sum of H p atoms as in (2.6), which converges in H p norm. Moreover, !1/p X p |λk | ≤ c||f ||H p . k
older) space Lip (X, γ) Let X be an open subset of Rn . For 0 < γ < 1 the Lipschitz (H¨ consists of functions f for which there exists a constant A such that |f (x)| ≤ A almost everywhere and sup |f (x − y) − f (x)| ≤ A|y|γ x
holds for all y with x−y ∈ X. Minimal A satisfying these two inequalities can be taken to be the norm of f (cf. [58], [66], [16]). The Hardy space H 1 plays an important role in the complex interpolation method. Proposition 2.6 Let Tz be a family of linear operators on Rn , parameterized by complex z with 0 ≤ Re (z) ≤ 1. Suppose that for all simple (step-) functions f, g, vanishing outside a set of finite measure, the map Z z 7→ (Tz f )gdx Rn
is bounded and analytic in the open strip 0 < Re (z) < 1 and is continuous in its closure. Suppose that ||Tz f ||L1 ≤ C0 ||f ||H 1 for Re (z) = 0 and ||Tz f ||q1 ≤ C1 ||f ||p1 for Re (z) = 1. Then also ||Tt f ||qt ≤ C01−t C1t ||f ||pt
(2.7)
with pt and qt defined by 1/pt = (1 − t) + t/p1 and 1/qt = (1 − t) + t/q1 .
The proof is based on the duality between H 1 and BM O ([25]).
Proposition 2.7 (Hardy-Littlwood-Sobolev) For every 0 < γ < n, 1 < p < q < ∞ and 1/q = 1/p − (n − γ)/n, there exists a constant Apq such that ||f ∗ (|y|−γ )||Lq ≤ Apq ||f ||Lp .
There is a similar result in Hardy spaces ([60, III.5.21]). Proposition 2.8 The operator Iα f =R f ∗ (|y|−γ ) allows an analytic extension on the set −n(p−1 − 1) ≤ Re γ < n, when xα f (x)dx = 0 hold for |α| ≤ n(p−1 − 1) and p ≤ 1. For every 0 < p < q < ∞ and 1/q = 1/p − (n − γ)/n, there exists a constant Apq such that ||f ∗ (|y|−γ )||H q ≤ Apq ||f ||H p .
The same result holds for q = ∞, p ≤ 1 and 0 < p = q < ∞, Re γ = n.
The development of the complex Hardy space theory can be traced in [68] for C and in [23], [58], [62] for Cn . The real theory in terms of maximal operators and CalderonZygmund decomposition can be found in [25]. Applications of Hardy spaces to several problems of the theory of singular integral operators appeared already in [30]. The general theory and applications can be found in [58], [62] and [60]. 18
2.3
Estimates
Let X and Y be smooth paracompact real manifolds of dimension n. Let T ∈ I μ (X, Y ; Λ) be a Fourier integral operator of order μ with the wave front equal to Λ0 = {(x, ξ, y, η) : (x, ξ, y, −η) ∈ Λ} ⊂ T ∗ X × T ∗ Y. The canonical relation Λ is a Lagrangian submanifold of T ∗ X × T ∗ Y equipped with the symplectic form σX ⊕ −σY . We will consider operators T , for which the canonical relation Λ locally is the graph of a symplectomorphism between T ∗ X\0 and T ∗ Y \0, equipped with canonical symplectic forms σX and σY . Let πX×Y be the canonical projection from T ∗ (X ×Y ) to X ×Y . By the equivalenceof-phase-function theorem, the integral kernel of the operator T can always be assumed to be a Fourier integral distribution of the form Z K(x, y) = eiΦ(x,y,θ) a(x, y, θ)dθ. (2.8) RN
The symbol a of the operator T has order m, a ∈ S m , where m = μ − (N − n)/2. By the equivalence-of-phase-function theorem (Theorem 2.3), the number N of frequency variables can be chosen to be equal to n and in this case the orders of the operator and its symbol coincide. The set Λ0 locally has the form Λ0 = {(x, ∇x Φ, y, ∇y Φ) : ∇θ Φ(x, y, θ) = 0}. Note, that the homogeneity of the canonical relation implies rank dπX×Y |Λ ≤ 2n − 1. The following theorem ([56]) is an important result on Lp continuity of operators from I μ. Theorem 2.9 Let Λ ⊂ T ∗ X\0 × T ∗ Y \0 be a conic Lagrangian manifold. Assume that Λ is a local canonical graph. Let 1 < p < ∞ and μ ≤ −(n − 1)|1/p − 1/2|. Then T ∈ I μ (X, Y ; Λ) is continuous from Lpcomp to Lploc . The order μ is sharp for elliptic operators T , provided that dπX×Y |Λ has the maximal rank somewhere, equal to 2n − 1. However, if the rank of dπX×Y |Λ does not attain 2n − 1, the order μ is not sharp and it depends on the singularities of the projection πX×Y |Λ . It turns out that the regularity properties of Fourier integral operators in Lp spaces with p 6= 2 depend on the maximal rank of the projection πX×Y restricted to Λ. An important ingredient is the following smooth factorization condition for πX×Y . Suppose that there exists a number k, 0 ≤ k ≤ n − 1, such that for every λ0 = (x0 , ξ0 , y0 , η0 ) ∈ Λ, there exist a conic neighborhood Uλ0 ⊂ Λ of λ0 and a smooth homogeneous of zero order map πλ0 : Λ ∩ Uλ0 → Λ with constant rank rank dπλ0 = n + k, for which holds πX×Y = πX×Y ◦ πλ0 .
(2.9)
Iρμ (X, Y
; Λ) with 1/2 ≤ ρ ≤ 1 have better Under this assumption, operators T ∈ continuity properties than those described in Theorem 2.9. Theorem 2.10 Let Λ ⊂ T ∗ X\0 × T ∗ Y \0 be a conic Lagrangian manifold satisfying the local graph condition. Let μ ≤ −(k + (n − k)(1 − ρ))|1/p − 1/2| with 1 < p < ∞ and 1/2 ≤ ρ ≤ 1. Let the smooth factorization condition (2.9) hold. Then operators T ∈ Iρμ (X, Y ; Λ) are continuous from Lpcomp to Lploc . 19
Note that the essential homogeneity of the symbol a is not important here. With m = μ − (N − n)/2 we get a ∈ Sρm , which means that α |∂x,y ∂θβ a(x, y, θ)| ≤ Cαβ (1 + |θ|)m−ρ|β|+(1−ρ)|α| .
Results on L2 continuity of such operators can be found in [33] for 1/2 < ρ and in [27] for ρ = 1/2. Operators T ∈ Iρ0 are continuous in L2 and this result does not depend on condition (2.9). Proofs of Theorems 2.9 and 2.10 are based on the complex interpolation method (as in Proposition 2.6) applied to the continuity in L2 and in the Hardy space H 1 , for which we have Theorem 2.11 Let Λ satisfy conditions of Theorem 2.10 and let the number N of frequency variables be equal to n. Let T be a Fourier integral operator with the Schwartz integral kernel K, given by the Lagrangian distribution (2.8). Assume that the symbol k/2+(n−k)(1−ρ)/2 in (2.8) vanishes outside a compact set in Rn x × Rn y . Then T a ∈ Sρ 1 is continuous from H (Rn ) to L1 (Rn ). Let us indicate briefly some ideas behind the proof of the above theorems. Let us begin with Theorems 2.9 and 2.10. Let a be a symbol of order −(n − 1)|1/p − 1/2| and we assume that 1 < p ≤ 2. Let Tz be a family of Fourier integral operators with Lagrangian distributions Z 1 Kz (x, y) = eiΦ(x,y,θ) a(x, y, θ)(1 + |θ|2 ) 2 [(n−1)(1/p−1/2)−(n−1)/2+z(n−1)/2] dθ. Rn
The continuity of zero order operators in L2 implies ||Tz f ||L2 ≤ Cz ||f ||L2 , Re z = 1, and Theorem 2.11 implies ||Tz f ||L1 ≤ Cz ||f ||H 1 , Re z = 0. Constants Cz depend on finitely many derivatives of the symbol and, therefore, have polynomial growth in |z|. Let us apply Proposition 2.6 with t = 2(1 − 1/p). We have 1/pt = 1/qt = 1 − t/2, (n − 1)(1/p − 1/2) − (n − 1)/2 + t(n − 1)/2 = 0, so that Tt = T and Theorem 2.9 follows from Proposition 2.6. Similarly, Theorem 2.10 follows from −(k+(n−k)(1−ρ))|1/p−1/2| Theorem 2.11 if we take a ∈ Sρ and kernels R Kz (x, y) = Rn eiΦ(x,y,θ) a(x, y, θ) 1 (1 + |θ|2 ) 2 [−(k+(n−k)(1−ρ))|1/p−1/2|+(z−1)[k+(n−k)(1−ρ)]/2] dθ. P To prove Theorem 2.11, it is convenient to use an atomic decomposition f = k λk ak , as in Proposition 2.5. If a ball B from the definition of an atom contains the unit cube, the Cauchy–Schwartz inequality implies ||T ak ||L1 ≤ C| supp ak |1/2 ||T ak ||L2 ≤ C 0 | supp ak |1/2 ||ak ||L2 , which is uniformly bounded in view of property (2) in the definition of atoms. Therefore, Proposition 2.5 reduces the statement of Theorem 2.11 to proving the uniform 20
bound ||T ak ||L1 ≤ C for all atoms supported in sufficiently small balls. This estimate can be obtained using a number of decompositions in the frequency space. First one uses a conic decomposition of the phase space and applies a dyadic decomposition to each cone. Using the corresponding partition of unity, it is possible to replace the phase function of T by a linear function, in each domain, so that integration by parts yields L1 estimates. It turns out that the error coming from such replacement is summable and, therefore, Theorem 2.11 follows. The details of this proof can be found in [56]. In Section 7 we will discuss sharpness of the statement of Theorem 2.10. In particular, using the stationary phase method we will show that for ρ = 1 the order μ in Theorem 2.10 is sharp for elliptic operators T , for which the rank n + k is attained somewhere. In Section 2.4 we will show that the factorization condition for πX×Y |Λ is equivalent to the factorization of the Hessian of a phase function. This leads in turn to a parametric Monge–Amp`ere equation (cf. [34]). This will be discussed in Section 6.2. In order to obtain the continuity from Lp to Lq one often uses the complex interpolation method again (Proposition 2.6). Let a ∈ S μ (Rn × Rn × Rn ) and let Φ(x, y, ξ) be a nondegenerate phase function. Let a Fourier integral operator be microlocally given by Z Z eiΦ(x,y,ξ) a(x, y, ξ)f (y) dy dξ. T f (x) = Rn
Rn
Theorem 2.12 Let T ∈ I μ and let 1 < p, q < ∞. The following holds: (1) Operator T : Lp (Rn ) → Lq (Rn ) is continuous provided μ ≤ n(1/q − 1/p) and p ≤ 2 ≤ q. (2) Operator T : Lp (Rn ) → Lq (Rn ) is continuous, provided μ ≤ 1/q − n/p + (n − 1)/2 and 1 < p ≤ q ≤ 2. The dual result holds for 2 ≤ p ≤ q < ∞. (3) For p = 1 the statements of items (1) and (2) hold if we replace L1 (Rn ) by the Hardy space H 1 (Rn ). 2 Remark 2.13 In the case rank Dξξ Φ = n−1 the statement of Theorem 2.12, (1), can be improved. Let 1 < p, q < ∞ and let p0 , q 0 denote their conjugates. Then operator T is continuous from Lp (Rn ) to Lq (Rn ) provided
1 n n−1 − + , q p 2 n 1 n−1 μ≤ − − , q p 2 μ≤
q ≤ p0 , q ≥ p0 ;
The statement extends to the case of p = 1 if we replace L1 (Rn ) by the Hardy space H 1 (Rn ). The proof of Remark 2.13 follows by the complex interpolation method between Theorem 2.9 and the fact that under conditions of Remark 2.13 the operator T is continuous from H 1 (Rn ) to L∞ (Rn ) provided μ = −(n + 1)/2. The sharpness of the bound for μ in Remark 2.13 is shown in [60, IX.6.16] for operators with the phase function Φ(x, y, ξ) = hx − y, ξi + |ξ|. See also Section 7. 21
Remark 2.14 It follows from Proposition 2.8 with p = 1 and q = 2 that operators of order −n/2 are continuous from H 1 to L2 . Proposition 2.15 Let the factorization condition (2.9) hold and let 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞, and μ ≤ −n/p + (n − k)/q + k/2. Then operators T ∈ I μ (X, Y ; Λ) are continuous from Lpcomp (Y ) to Lqloc (X). The statement follows from the interpolation method between the statements of Theorem 2.10 and Remark 2.14. Similarly, we get that operators from I μ are continuous from Lip (α − k/2 − μ) to Lip (α) for all α ∈ R. As a consequence, we get a statement for pseudo-differential operators. Proposition 2.16 Let P ∈ Ψμ (X) and let 1 < p ≤ q < ∞. Then P : Lpcomp (X) → Lqloc (X) is continuous, when μ ≤ −n(1/p − 1/q). Proof. Let us give a direct proof without using the interpolation technique. The case p = q is well known and it follows, for example, from Theorem 2.10 with k = 0 and μ ≤ 0. Let us assume now that p < q. The operator (I −Δ)−μ/2 ◦P is continuous in Lp and the statement reduces to the properties of the operator (I − Δ)μ/2 . Its principal symbol is homogeneous of degree μ and its integral kernel has degree −n − μ in |x − y|. Therefore, the operator (I − Δ)μ/2 is of the form Z |x − y|−γ f (y) dy Iγ f (x) = Rn
with γ = n + μ. For μ < 0, it follows from Proposition 2.7 that Iγ is continuous from Lp to Lq provided 1/q = 1/p + μ/n. This completes the proof of Proposition 2.16. Let us call a Fourier integral operator analytic if it acts in real analytic manifolds X and Y , and if its canonical relation is analytic. For such operators, the phase function is analytic and the smooth factorization condition can be extended to the complex domain. Further analysis of this condition can be applied to the Lp continuity of operators in Rn with n ≤ 4, and in Rn , with an arbitrary n and additional condition rank dπX×Y |Λ ≤ n + 2. This will be discussed in Section 5.
2.4
Parametric fibrations
First let us reduce the general manifold setting to open sets in Rn . It does not restrict the generality since we are interested in local properties of operators. Note that sets X and Y have equal dimensions which follows from the condition that the canonical relation is a local graph. In general, the canonical relation of a Fourier integral operator in T ∗ X ×T ∗ Y can be regarded as a smooth family of Lagrangian submanifolds of T ∗ Rn parameterized by points in T ∗ Rn . Let us first show that from this point of view the ranks of the projection to the base space differ by n. It follows from the equivalenceof-phase-function theorem (Theorem 2.3) that a homogeneous canonical relation has the form Λ = {(∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ))} (2.10) in a neighborhood of a point (x0 , ξ0 , y0 , η0 ). The generating function φ satisfies ∇ξ φ(y0 , ξ0 ) = x0 , ∇y φ(y0 , ξ0 ) = η0 . 22
The phase function of an operator T ∈ Iρμ (X, Y ; Λ) has the form Φ(x, y, ξ) = hx, ξi − φ(y, ξ)
(2.11)
and Λ = ΛΦ is locally the collection of the points {(∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ))}. The set in (2.10) will be often denoted by ΛΦ , with phase function Φ as in (2.11). Microlocally, the Schwartz kernel (2.8) of operator T has the form Z ei(hx,ξi−φ(y,ξ)) b(x, y, ξ)dξ (2.12) K(x, y) = Rn
with some symbol b ∈ Sρμ . Therefore, locally, modulo smooth terms, the Schwartz kernels of operators in Iρμ (X, Y ; Λ) are finite sums of kernels of the form (2.12) with symbols b ∈ Sρμ . Each of the kernels has this form in its own coordinate system. Therefore, we will assume that X and Y are open sets in Rn . The local graph condition is equivalent to the condition det φ00yξ (y, ξ) 6= 0
(2.13)
on the support of the symbol of T . In this local coordinate system, the projection πX×Y |Λ takes the form πX×Y |Λ : (∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ)) 7→ (∇ξ φ(y, ξ), y).
(2.14)
The level sets of the projection πX×Y |Λ can be parameterized by points ξ, for which the gradient ∇ξ φ(y, ξ) is constant. On the other hand, in view of Proposition 2.1 an open submanifold Λ0 of the canonical relation Λ is equal to the conormal bundle N ∗ Σ0 of a smooth submanifold Σ0 of the set Σ = πX×Y (Λ). It follows that the level set of the projection πX×Y |Λ in Λ0 at the point (x0 , ξ0 , y0 , η0 ) is a linear subspace ∗ of T(x (X × Y ). By (2.14), these linear subspaces are parameterized by the pairs 0 ,y0 ) (ξ, ∇y φ(y, ξ)). Therefore, points ξ in the level set of ∇ξ φ(y, ξ) form a linear subspace of the n–dimensional frequency space Ξ. It is clear how this subspace depends on (y, ξ): Lemma 2.17 The mapping γ : Y × Ξ → T ∗ X × T ∗ Y defined by γ(y, ξ) = (∇ξ φ(y, ξ), ξ, y, ∇y φ(y, ξ)) is a diffeomorphism from Y × Ξ to Λ. For every y ∈ Y , the restriction ξ 7→ γ(y, ξ) is a diffeomorphism from Ξ to Λ ∩ (Rn × Rn × {y} × Rn ), with the inverse given by the projection (x, ξ, y, η) 7→ ξ. Therefore, the linear mapping dπX×Y |Λ is isomorphic to dπX×Y ◦ dγ|Y ×Ξ and, in particular, their kernels are isomorphic. The latter is ker dπX×Y ◦ dγ|Y ×Ξ (y, ξ) = {(δy, δξ) :
∂2φ ∂2φ (y, ξ)δξ + (y, ξ)δy = 0, δy = 0}. 2 ∂ξ ∂y∂ξ
We obtain ker dπX×Y ◦ dγ|Y ×Ξ (y, ξ) = (0, ker
∂2φ (y, ξ)). ∂ξ 2
(2.15)
We proved the following characterization of the projection in terms of the phase function: 23
Theorem 2.18 Let a local graph Λ be defined by the generating phase function Φ(x, y, ξ) = hx, ξi − φ(y, ξ). Then, for every 0 ≤ k ≤ n − 1, the following statements are equivalent. (1) rank dπX×Y |ΛΦ ≤ n + k. (2) rank dπX×Y ◦ dγ|{y}×Ξ ≤ k for all y ∈ Y and γ from Lemma 2.17. (3) rank
∂2φ ∂ξ 2 (y, ξ)
≤ k for all y ∈ Y and ξ ∈ Ξ.
Remark 2.19 By the linearity of the level sets of the projection πX×Y |Λ , it follows that the equality −1 πX×Y (πX×Y (λ)) ∩ Λ = ker dπX×Y |Λ (λ) holds locally at every λ ∈ Λ. On the other hand, if we use linearity of the level sets of the mapping ∇ξ φ(y, ξ) and the isomorphism in (2.15), we get that the level sets of the mapping ξ 7→ ∇ξ φ(y, ξ)
2 are defined by the kernels of the matrix Dξξ φ(y, ξ). The same conclusion follows if we apply the argument of Section 4 to the function φ directly.
The level sets of πX×Y |Λ correspond to the level sets of the gradient ∇ξ φ(y, ξ) with respect to ξ. In particular, they are disjoint and are extendible to an open set Ω in Cn , provided that φ is analytic. Therefore, we have fibrations in Ω ∩ Rn and Ω by the level sets (with respect to ξ) of the gradient ∇ξ φ(y, ξ) and its holomorphic extension. Remark 2.20 If the maximal rank of dπX×Y |Λ equals n + k, we denote by Λ(k) the set on which it is attained. In terms of fibrations, the fibration by linear subspaces in Remark 2.19 is also defined by the smooth mapping κR : γ −1 (Λ(k) ) 3 (y, ξ) 7→ ker
∂2φ (y, ξ) ∈ Gn−k (Rn ). ∂ξ 2
(2.16)
The factorization condition (2.9) is equivalent to the condition that the mapping κR ◦ γ −1 is locally smoothly extendible from Λ(k) to Λ. This extension corresponds to the fibration by the kernels ker dπλ0 , where πλ0 is as in (2.9). In terms of fibrations, the set of essential singularities of the mapping κR is equal to Ωsing ∩ (Rn × Rn ), where Ωsing is the set of essential singularities of the complex extension of the mapping κR . The following example shows that the factorization condition is not trivial. In view of Remarks 2.19 and 2.20, by Theorem 2.18 the factorization condition reduces to the study of the inequality ker φ00ξξ (y, ξ) ≤ k. In this case we have rank dπX×Y |ΛΦ ≤ n + k by Theorem 2.18, (3). The function φ(y, ξ) = hy, ξi +
k+1 1 X (y1 ξ1 + yi ξi )2 ξn i=2
24
satisfies the necessary rank condition in a neighborhood of ξn = 1. On the other hand, we have Pk+1 y1 i=2 (y1 ξ1 + yi ξi ) y2 (y1 ξ1 + y2 ξ2 ) .. . 2 yk−1 (y1 ξ1 + yk−1 ξk−1 ) ∇ξ φ(y, ξ) = y + (2.17) . 0 ξn . .. 0 P k+1 1 2 (y ξ + y ξ ) 1 1 i i i=2 ξn
For points y with yi 6= 0, 1 ≤ i ≤ k + 1, the level sets in (2.17) correspond to the level sets if the mappings y1 ξ1 + yi ξi , 2 ≤ i ≤ k + 1. The direction of the level sets is now determined by fractions yi /y1 , 2 ≤ i ≤ k + 1, which are not continuous at y = 0. We obtain Example 2.21 Let 1 ≤ k ≤ n − 2 and let x, y, ξ ∈ Rn . The function Φ(x, y, ξ) = hx − y, ξi − satisfies the rank condition
k+1 1 X (y1 ξ1 + yi ξi )2 ξn i=2
rank dπX×Y |ΛΦ ≤ n + k and defines a local canonical graph ΛΦ , for which the fibration by the level sets of the mapping πX×Y |ΛΦ does not allow a continuous extension over y = 0. Remark 2.22 In the case of k = 0, the operators are conormal and their analysis can be reduced to the analysis of pseudo-differential operators by composing them with Fourier integral operators induced by a smooth coordinate transformation (see Section 7.2). For such operators, the factorization condition is trivially satisfied. The case of k = n − 1 correspond to the inequality rank dπX×Y |Λ ≤ 2n − 1, when the factorization condition is satisfied because Λ is conic. In this case, one can take πλ0 in (2.9) to be the projection in the conic direction. Remark 2.23 The failure of the factorization condition implies that the singular support Σ of an operator T can not be a smooth manifold. Indeed, if Σ is a smooth manifold, then Λ = N ∗ Σ and the factorization condition holds. Conversely, the fact that Σ is not smooth does not imply the failure of the factorization condition. For example, let us take a semicubic parabola G in R3 : G = {x ∈ R3 : x21 = x32 , x3 = 0}
and define Σ by Σ = {(x, y) ∈ R6 : x−y ∈ G}. It is straightforward that a corresponding phase function Φ has the form Φ(x, y, ξ) = hx − y, ξi −
4 ξ23 . 27 ξ12
It is clear that Σ is not smooth at zero. Meanwhile, the factorization condition holds for Φ, which will become clear in Section 5.3. 25
In Section 4, we will discuss fibrations in the real setting in the classes C m . For now, we restrict ourselves to one example. Example 2.24 For k ∈ N the function Φ(x, y, ξ) = hx − y, ξi −
1 k+1 k+1 (y y2 ξ1 + (y12 + y22 )ξ2 )2 , y ∈ Rn , xi ∈ Rn ξn 1
satisfies condition rankDξ2 Φ ≤ 1 and defines a local canonical graph, for which the fibration by level sets of the function ∇ξ Φ is continuously extendible over y = 0. Moreover, this extension is locally C 2k−1 , but not C 2k .
3 3.1
Affine fibrations Methods of complex analytic geometry
In this section, we will review some notions and facts of complex analytic geometry which will be frequently used in the sequel. Following [49], let us define meromorphic mappings. Definition 3.1 A mapping τ : X → Y between complex manifolds X and Y is called meromorphic, if the following three conditions hold. (1) For every x ∈ X the image set τ (x) ⊂ Y is non-empty and compact in Y . (2) The graph of the mapping τ , that is the set all pairs (x, y) ∈ X × Y such that y ∈ τ (x), is a connected complex analytic subset of X × Y of dimension equal to the dimension of X. (3) There exist a dense subset X ∗ of X, such that for every x ∈ X ∗ the image set τ (x) consists of a single point. One of the important tools for the analysis of the structure of analytic sets are dimension estimates. Let M be a complex n-dimensional manifold and let E ⊂ M be an arbitrary subset of M . Dimension of E is defined by dim E = sup{dim F : F ⊂ E}, where supremum is taken over all smooth submanifolds of M contained in E. If the set on the right hand side is empty, we assume that the supremum is −∞. If E is a submanifold or an analytic subset of M , then this notion of dimension coincides with the standard, used in the analytic geometry. If π is a projection from the Cartesian product to one of the sets, then the upper bound for the dimension of the preimage is given by the following theorem. Theorem 3.2 Let E ⊂ M × N , where M, N are complex manifolds. Let π : E → M be the natural projection. Assume that for k ∈ N holds dim π −1 (z) ≤ k, ∀z ∈ π(E). Then dim E ≤ k + dim π(E). 26
By a globally analytic subset of the manifold M we mean any set of the form Z(f1 , . . . , fk ) = {z ∈ M : f1 (z) = . . . = fk (z) = 0}, where f1 , . . . , fk are some holomorphic functions on M . A subset Z of the manifold M is called analytic if every point of M has an open neighborhood U such that the set Z ∩ U is globally analytic in U . Analytic subsets of open sets in M are called locally analytic in M . In particular, a set Z is analytic in M if and only if it is locally analytic and closed. Proposition 3.3 (1) Let V ⊂ M and W ⊂ N be non-empty subsets of manifolds M and N , respectively. Then the product V × W is (locally) analytic in M × N if and only if both V and W are (locally) analytic in M and N , respectively. (2) (The Analytic Graph Theorem) Every continuous mapping f : M → N with analytic graph is holomorphic. (3) Every proper analytic subset Z of a connected manifold M is nowhere dense and its complement M \Z is open and connected. If V ⊂ M and W ⊂ N are locally analytic subsets, then the mapping f : V → W is called holomorphic if every point in V has an open neighborhood U in M such that f |U ∩V is the restriction of a holomorphic mapping from U to N . Theorem 3.4 Let k ∈ N and let f be a holomorphic mapping defined above. Then (1) dim V ≥ k + dim f (V ), if dim f −1 (w) ≥ k for all w ∈ f (V ). (2) dim V ≤ k + dim f (V ), if dim f −1 (w) ≤ k for all w ∈ f (V ). The proof of the first part is based on the reduction of the estimate to a subset Z of the image f (V ) of dimension equal to dim f (V ). So, in this case, the set f (V ) can be semi-analytic. The second estimate is based on Theorem 3.2. If the level sets are of the same dimension, we have Corollary 3.5 Let Z ⊂ W be a locally analytic subset of N . If dim f −1 (w) = k holds for all w ∈ Z, then dim f −1 (Z) = k + dim Z. For z ∈ V let lz f denote the germ of the fiber of f lz f = f −1 (f (z)) at the point z.
z
Theorem 3.6 (1) (semicontinuity) The mapping V 3 z 7→ dim lz f is upper semicontinuous: for all a ∈ V the inequality dim lz f ≤ dim la f holds in some neighborhood of a. (2) The inequality dim lz f ≥ dimz V − dim f (V ) holds for all z ∈ V . (3) (Cartan–Remmert). For every k ∈ N the set {z ∈ V : dim lz f ≥ k} is analytic.
27
The set E is called analytically constructible if there exist analytic sets V and W such that E = V \W . Theorem 3.7 Let V, W ⊂ M be analytic sets. Then the closures of any connected component of the set V \W , any open and closed set in V \W , and, in particular, the set V \W , are unions of some simple components of V not contained in W , and so they are analytic. The family of the connected components of the set V \W is locally finite. Proofs of the above statement can be found in [35] in II.1.4, II.3.4, V.1.1, II.3.6, V.3.2, V.3, IV.2.10. By [49] and [35, V.5.1] we have: Theorem 3.8 (Remmert’s Proper Mapping Theorem) If f : X → Y is a proper holomorphic mapping of analytic spaces, then its range f (X) is analytic in Y . Consequently, the image of each analytic subset of X is an analytic subset of Y .
3.2
Main results
Theorem 3.9 Let Γ satisfy conditions (A1)–(A3). Assume that the set Ωsing is not empty. Then the following holds. (1) The set of essential singularities Ωsing is an analytic subset of Ω, and for all ξ ∈ Ωsing holds max{k − 1, n − k + 1} ≤ dimξ Ωsing ≤ n − 2. In particular, 3 ≤ k ≤ n − 1 and n ≥ 4. (2) Let ξ ∈ Ωsing be a regular (smooth) point, let ξ = limi→∞ ξi , ξi ∈ Ω(k) and let Gn−k (Cn ) 3 κ = limi→∞ κ(ξi ). Then κ ⊂ Tξ Ωsing . As a corollary we get that all singularities are removable when k ≤ 2. Theorem 3.10 Let Γ satisfy conditions (A1)–(A3) with k ≤ 2. Then the set of essential singularities Ωsing is empty and κ has a regular (holomorphic) extension to Ω. Estimates for the dimension of the set Ωsing in Theorems 3.9, 3.10 are sharp in the sense that for all intermediate dimensions there exists a mapping Γ, for which this intermediate dimension of the singular set is attained. Theorem 3.11 Let k, p, d be any natural numbers such that 3 ≤ k ≤ n − 1, p ≥ k, and 2 ≤ d ≤ min{k − 1, n − k + 1}. Then there exists a holomorphic mapping Γ : Cn → Cp , for which conditions (A1)–(A3) are satisfied and dim Ωsing = n − d. Moreover, there exists Γ for which also Ω\Ω(k) = Ωsing . All statements remain true if we replace Cp in the definition of Γ by an arbitrary analytic space. The relation between properties (A1)–(A3) and fibrations by affine level sets is as follows: Theorem 3.12 Let a holomorphic mapping Γ : Ω → Cp be defined on an open set Ω ⊂ Cn . Let the set Ω0 be open and dense in Ω and assume that for all ξ ∈ Ω0 the level sets Γ−1 (Γ(ξ)) through ξ are affine subspaces of Ω of codimension k with k < n. Then Γ satisfies conditions (A1)–(A3) and Ωsing ⊂ Ω\Ω0 . 28
Under conditions (R1)–(R3) we have: Theorem 3.13 Let Γ satisfy conditions (R1)–(R3) with k = 1 and let (y, ξ) ∈ Ωsing . Then the mapping η 7→ Γ(y, η) is constant. Let us first prove Theorem 3.11. Since k − 1 ≤ n − d, d ≤ n/2 and d ≤ k − 1 ≤ n − d, the mapping Γ : (x1 , . . . , xd , y1 , . . . , yn−d ) 7→ (
d X
yi xi , y1 , . . . , yk−1 , 0),
i=1
with 0 ∈ Cn−k+2 is well defined. If not all yi with 1 ≤ i ≤ d are equal to zero at a point, then rank DΓ at this point equals k. On the other hand it is clear that points with yi = 0 for all 1 ≤ i ≤ d belong to Ωsing . Next let us prove Theorem 3.13. Lemma 3.14 Let Γ satisfy conditions (R1) and (R2). For ω ∈ Ωsing and for every k-dimensional linear subspace C of Cn there exists a sequence ωj ∈ Ω(k) such that ωj converges to ω as j → ∞, κ(ωj ) converges to κ ∈ Gn−k (Cn ) as j → ∞, and κ ∩ C 6= {0}. Proof. The set G(C) = {L ∈ Gn−k (Cn ) : L ∩ C = {0}} is holomorphically diffeomorphic to Ck(n−k) (cf. [35, B.6.6] and [35, Prop., p.367]). Therefore, if there exist a neighborhood U of the point ω in Ω such that the image κ(U ∩ Ω(k) ) is contained in a compact set in G(C), then ω 6∈ Ωsing . Lemma 3.15 Let Γ satisfy conditions (R1)–(R3). For every point (y, ξ) ∈ Ωsing and for every k-dimensional linear subspace C of Cn there exists a linear subspace L of C with dim L ≥ 1 such that for every l ∈ L holds Γ(y, ξ + l) = Γ(y, ξ).
Proof. According to Lemma 3.14 there exists a sequence ωj → (y, ξ) with ωj ∈ Ω(k) such that the limit limj κ(ωj ) = κ exists and κ ∩ C 6= {0}. Let L = κ ∩ C. In view of condition (R3) for all ωj = (yj , ξj ) and zj ∈ κ(ωj ) holds Γ(yj , ξj + zj ) = Γ(yj , ξj ). The statement of the lemma now follows from the continuity of Γ. The proof of Theorem 3.13 follows from Lemma 3.15 with k = 1, since in this case Lemma 3.15 holds for an arbitrary C. Corollary 3.16 Let Γ satisfy (A1)–(A3) with k = 1. Then the set of essential singularities Ωsing is empty. Proof. Let ξ ∈ Ωsing . According to Theorem 3.13 the mapping η 7→ Γ(η) is constant in Ω and the matrix Dξ Γ vanishes. This contradicts k = 1. Let us give an example of a singular fibration, dependent of two parameters, for arbitrary 1 ≤ k ≤ n − 1.
29
Example 3.17 For 1 ≤ k ≤ n − 1, m ≥ 2, define y1 ξ1 + y2 ξ2 .. k Γ(y, ξ) = ∈C , . y1 ξ1 + y2 ξk+1
where y ∈ Cm and ξ ∈ Cn . Then the maximal rank y1 y 2 0 ∙ ∙ ∙ 0 y1 0 y2 ∙ ∙ ∙ 0 Dξ Γ(y, ξ) = . .. .. .. .. .. . . . . y 1 0 0 ∙ ∙ ∙ y2 equals k, and for all y with y1 y2 = 6 0 1 −y1 /y2 .. . −y1 /y2 1 0 .. . 0
of the matrix ∙∙∙ 0 ∙∙∙ 0 k×n .. .. ∈ C . . ∙∙∙ 0
the kernel κ(y, ξ) is spanned by the vectors 1 −y1 /y2 .. . −y1 /y2 ,∙∙∙ , . 0 . .. 0 1
Consequently, the set of essential singularities is Ωsing = {(y, ξ) ∈ Cm × Cn : y1 = y2 = 0}. Let us continue our analysis of the set Ωsing of essential singularities. In the sequel we will always assume that the mapping Γ satisfies conditions (A1), (A2) and (A3). It should be noted at once that condition (A3) guaranteeing the linearly of the level sets of Γ is essential for the analysis. Conditions (A1), (A2) alone do not imply the statement of Corollary 3.16. For example, for the mapping Γ(ξ) = (ξ12 + ξ22 + ξ32 , 0, 0) ∈ C3 it is not difficult to check that k = 1 and Ωsing = {0}. However, in this case Γ is not constant on ξ + κ(ξ) and (A3) is not satisfied. Condition (A3) can be characterized in the following way: Proposition 3.18 For every ξ ∈ Ω(k) the mapping Γ is constant on (ξ + κ(ξ)) ∩ Ω, i.e. Γ(ζ) = Γ(ξ) (3.1) for all ζ ∈ (ξ + κ(ξ)) ∩ Ω, if and only if κ is constant on (ξ + κ(ξ)) ∩ Ω, i.e. κ(ξ) ⊂ ker DΓ(ζ)
(3.2)
for all ζ ∈ (ξ + κ(ξ)) ∩ Ω. The equality in (3.2) holds for ζ ∈ (ξ + κ(ξ)) ∩ Ω(k) . Proof. Differentiation of (3.1) with respect to the basis vectors of κ(ξ) implies (3.2). Indeed, let ej (ξ), j = 1, . . . , n − k, be a basis of vectors spanning the linear space κ(ξ), locally in Ω(k) . Then (3.1) implies that Γ(ξ + tej (ξ)) = Γ(ξ), j = 1, . . . , n − k, 30
(3.3)
for small t and ζ = ξ + tej (ξ). Differentiation of (3.3) at t = 0 implies DΓ(ξ + tej (ξ))ej (ξ) = 0,
(3.4)
which, in turn, implies (3.2). For ζ ∈ Ω(k) , dimensions of both sides coincide and the equality holds. Conversely, it follows from (3.2) that DΓ(ξ + η)κ(ξ) = 0 for η ∈ κ(ξ) with ξ + η ∈ Ω. Hence, Γ(ξ + η) is constant on κ(ξ), i.e. equality (3.1) holds. Indeed, for ζ = ξ + tej (ξ) equality (3.4) can be expressed as ∂Γ/∂t(ξ + tej (ξ)) = 0, which implies (3.1). Because Proposition 3.18 holds locally in a neighborhood of any point ξ ∈ Ω(k) and the mapping Γ is holomorphic, it also holds globally in Ω. Remark 3.19 Let us note without proof what happens in the case when rank drops by one. Let ξ ∈ Ωsing and ξ ∈ Ω(k−1) , i.e. rank DΓ(ξ) = k − 1. Then Γ(ξ + λ) = Γ(ξ) for all λ ∈ ker DΓ(ξ) with ξ + λ ∈ Ω. The graph of the mapping κ has the form G = {(ξ, L) ∈ Ω × Gn−k (Cn ) : ξ ∈ Ω(k) , L = κ(ξ)}.
(3.5)
Let E = {(ξ, L) ∈ Ω × Gn−k (Cn ) : L ⊂ ker DΓ(ξ)}.
The set E is a closed analytic set in Ω × Gn−k (Cn ) and
G = (Ω(k) × Gn−k (Cn )) ∩ E = E\{(ξ, L) ∈ Ω × Gn−k (Cn ) : ξ ∈ Ω\Ω(k) }. Therefore, G is the complement in E to a closed analytic set E ∩ {(ξ, L) ∈ Ω × Gn−k (Cn ) : ξ ∈ Ω\Ω(k) }. It means that G is analytically constructible. Definition 3.20 Let κ(ξ) ˉ ⊂ Gn−k (Cn ) denote the set of all limit points of the sequence κ(ξj ) as ξj → ξ, ξj ∈ Ω(k) . For V ⊂ Gn−k (Cn ) let Ve be defined by Ve =
[
L.
L∈V
Let us list the main properties of the mapping V 7→ Ve :
Proposition 3.21 Let V be an analytic set in Gn−k (Cn ). Then Ve is an analytic set in Cn . Moreover, dim Ve ≤ dim V + n − k. If dim V ≥ 1, then dim Ve ≥ n − k + 1.
Proof. The set
I = {(ξ, L) ∈ Cn × Gn−k (Cn ) : ξ ∈ L}
is analytic in Cn ×Gn−k (Cn ). Let π1 : (ξ, L) 7→ ξ be the projection from Cn ×Gn−k (Cn ) to Cn . Let V be an analytic set in Gn−k (Cn ). Then Ve can be written as Ve = π1 (I ∩ (Cn × V )). 31
The set Cn ×V and, therefore, I ∩(Cn ×V ), is analytic in Cn ×Gn−k (Cn ). By Remmert theorem 3.8, Ve is analytic, because the projection π1 is proper. The latter follows from the compactness of the Grassmannian Gn−k (Cn ). Let us now prove the estimate for the dimension. Let π2 : (ξ, L) 7→ L be the projection from Cn × Gn−k (Cn ) to Gn−k (Cn ). We have π2 (I ∩ (Cn × V )) = V. Let v ∈ V ⊂ Gn−k (Cn ). The level set of π2 at the point v in I ∩ (Cn × V ) consists of points {(ξ, v) : ξ ∈ v} and g = n − k. dim(π2−1 (v) ∩ I ∩ (Cn × V )) = dim {v}
According to Corollary 3.5 we must have dim(I ∩ (Cn × V ) = n − k + dim V . Projection π1 can not increase the dimension, which implies the estimate in the proposition. Assume now that dim V ≥ 1 and dim Ve = n − k. The set Ve is analytic and hence it has a finite number of irreducible components at each point. By definition of Ve , each of these components can only be a (n − k)-dimensional subspace of Cn . But this would mean that the set V consists of finitely many isolated points, which would contradict to dim V ≥ 1. This completes the proof. Let us describe the main properties of the mapping κ. Proposition 3.22 (1) The graph G in (3.5) is analytically constructible, its closure ˉ is analytic. The set κ(ξ) G ˉ is analytic and connected for every ξ ∈ Ω. The set e κ(ξ) ˉ is analytic in Cn . e (2) For every ξ ∈ Ω holds κ(ξ) ˉ ⊂ ker DΓ(ξ).
e ˉ ≥ 1 and dim κ(ξ) ˉ ≥ n − k + 1. On the other hand, (3) If ξ ∈ Ωsing , then dim κ(ξ) if ξ ∈ Ω\Ωsing , then the set κ(ξ) ˉ consists of a single element L ∈ Gn−k (Cn ) and e κ(ξ) ˉ = L.
e (4) Moreover, if ξ ∈ Ωsing and if C is an irreducible component of the set κ(ξ), ˉ then e ˉ ≥ n−k+1. dim C ≥ n−k+1. In particular, for all ξ ∈ Ωsing holds dimξ (ξ + κ(ξ))
e e (5) If ξ ∈ Ω(k−1) ∩ Ωsing , then κ(ξ) ˉ = ker DΓ(ξ) and dim κ(ξ) ˉ = n − k + 1. e (6) For every ξ ∈ Ω the mapping Γ is constant (ξ + κ(ξ)) ˉ ∩ Ω.
Proof. (1) We have already shown above that G is analytically constructible. Its ˉ is analytic as the closure of an analytically constructible set (Theorem 3.7). closure G It follows that ˉ κ(ξ) ˉ = {L ∈ Gn−k (Cn ) : (ξ, L) ∈ G}
ˉ is an analytic set ˉ = ({ξ} × Gn−k (Cn )) ∩ G is analytic in Gn−k (Cn ). Indeed, {ξ} × κ(ξ) and κ(ξ) ˉ is analytic by Proposition 3.3, (1). Let us show now that κ ˉ (ξ) is connected. Let U, V be disjoint open sets in Gn−k (Cn ), U ∩ V = ∅, such that κ ˉ (ξ) ⊂ (U ∪ V ). Let A = {η ∈ Ω(k) : κ(η) ∈ U } and B = {η ∈ (k) Ω : κ(η) ∈ V }. Then A and B are disjoint open sets in Ω(k) . Hence there exists an open neighborhood W of the point ξ such that the set W ∩ Ω(k) is connected in W ∩ Ω(k) ⊂ A ∪ B. Hence either A ∩ W ∩ Ω(k) = ∅ or B ∩ W ∩ Ω(k) = ∅. It follows 32
that either κ ˉ (ξ) ∩ U = ∅ or κ(ξ) ˉ ∩ V = ∅. Thus, the set κ ˉ (ξ) is connected. The set e κ(ξ) ˉ is analytic in Cn in view of Proposition 3.21. (2) follows from the continuity of DΓ and Proposition 3.18. ˉ (ξ) consists of a single point and dim κ ˉ (ξ) = 0. Conversely, (3) If ξ 6∈ Ωsing , then κ if ξ ∈ Ωsing , then for every C ∈ Gk (Cn ) the set Gn−k (Cn )C = {L ∈ Gn−k (Cn ) : L ∩ C 6= {0}} is a hypersurface. Condition κ ˉ (ξ) ∩ Gn−k (Cn )C 6= ∅ for all C ∈ Gk (Cn ) (Lemma 3.14) implies that the set κ ˉ (ξ) is infinite and dim κ ˉ (ξ) > 0. It follows now from Proposition e 3.21 that dim κ(ξ) ˉ ≥ n − k + 1. ˉ The set κ ˉ (ξ) is connected (4) Let dim C ≤ n − k. Then C ∈ Gn−k (Cn ), C ∈ κ(ξ). by (1) and contains more than one element by (3). It follows that C belongs to the closure of the regular part of positive dimension of the set κ ˉ (ξ). Therefore, C is e contained in an irreducible component of dimension ≥ n − k + 1 of the set κ(ξ). ˉ This e contradicts the assumption that C is an irreducible component of the set κ(ξ). ˉ (5) If ξ ∈ Ω(k−1) ∩ Ωsing , then ker DΓ(ξ) is a linear subspace of Cn of codimension e k − 1. From ξ ∈ Ωsing and (2) it follows that κ(ξ) ˉ has codimension k − 1. Let us now prove the equality in (5). For η ∈ ker DΓ(ξ), η 6= 0, let L denote the line through η. Then L ⊂ ker DΓ(ξ) and there exists C ∈ Gk (Cn ) such that L = ker DΓ(ξ) ∩ C. According to Lemma 3.14 there exists κ0 ∈ κ(ξ) ˉ with dim(κ0 ∩ C) ≥ 1. Then κ0 ∩ C ⊂ ker DΓ(ξ) ∩ C = L and, therefore, κ0 ∩ C = L. Finally, it follows that e e η ∈ L = κ0 ∩ C ⊂ κ0 ⊂ κ(ξ) ˉ and κ(ξ) ˉ and ker DΓ(ξ) are equal. e ˉ be such that (6) For ξ ∈ Ω(k) statement (5) is equivalent to (A3). Let η ∈ κ(ξ) ξ + η ∈ Ω. Then there exists sequences ξj , ηj ∈ Ω(k) such that ηj ∈ κ(ξj ), ξj + ηj ∈ Ω, ξj → ξ i ηj → η. Condition (A3) means that Γ(ξj + ηj ) = Γ(ξj ) and by continuity of Γ we get that Γ(ξ + η) = Γ(ξ). This completes the proof of the theorem. Note that because the graph of the mapping κ ˉ is analytic in Ω × Gn−k (Cn ), by the analytic graph theorem (Proposition 3.3, (2)), it follows that κ ˉ is analytic if it is locally bounded. For k = n − 1 we have Gn−k (Cn ) = P(Cn ), and by Chow theorem ([18]) it follows that the set κ ˉ (ξ) is algebraic and can be represented as the zero set of a system of homogeneous polynomials in Cn . For other values of k one can use the Pl¨ ucker embedding Gn−k (Cn ) → P(Λn−k Cn ) and Chow theorem to show that κ ˉ (ξ) is algebraic in Λn−k Cn . Let us now prove Theorem 3.12. Proof of Theorem 3.12. Assume that conditions (A1) and (A2) are satisfied for Γ with some m ≤ n instead of k. By the implicit function theorem, the level sets of Γ have dimension n − m on the set where the rank of the Jacobian DΓ is maximal. Hence m = k and conditions (A1) and (A2) are satisfied with m = k. For ξ ∈ Ω0 define λ(ξ) = Γ−1 (Γ(ξ)) − ξ. By the assumption of the theorem λ is a mapping from Ω0 to Gn−k (Cn ). The mapping Γ is constant on (ξ + λ(ξ)) ∩ Ω and hence, as in Proposition 3.18, we have λ(ξ) ⊂ ker DΓ(ζ) for ζ ∈ (ξ + λ(ξ)) ∩ Ω. In particular, it follows that λ(ξ) = ker DΓ(ξ) for all ξ ∈ Ω0 ∩ Ω(k) . This means that condition (A3) is satisfied on Ω0 ∩ Ω(k) . The set Ω0 ∩ Ω(k) is dense in Ω(k) and in Ω. Therefore, for every ξ ∈ Ω(k) there exists a sequence ξj ∈ Ω0 ∩ Ω(k) , convergent to ξ. In view of the compactness of Gn−k (Cn ) we get a subsequence ξj such that the corresponding sequence κ(ξj ) converges to some κ0 ∈ Gn−k (Cn ). Without loss of generality we can denote this subsequence by ξj as well. For every η ∈ (ξ +κ0 )∩Ω there exists a sequence 33
ηj ∈ (ξj + κ(ξj )) ∩ Ω such that ηj converges to η. From the above proof of condition (A3) on Ω0 ∩ Ω(k) follows that Γ is constant on ξj + κ(ξj ) and Γ(ηj ) = Γ(ξj ). Letting j go to infinity, we get Γ(η) = Γ(ξ) and η ∈ Γ−1 (Γ(ξ)). Because the argument holds for all η ∈ (ξ + κ0 ) ∩ Ω, we get (ξ + κ0 ) ∩ Ω ⊂ Γ−1 (Γ(ξ)). By the implicit function theorem, the level set Γ−1 (Γ(ξ)) is a smooth analytic manifold in Ω of codimension k and, therefore, it coincides with ξ + κ0 locally in a neighborhood of ξ. On the other hand, κ is holomorphic at ξ. Hence κ(ξ) = κ0 and condition (A3) is satisfied for all ξ ∈ Ω(k) . Finally, if ξ ∈ Ωsing , then in view of Proposition 3.22, (6), the level set of Γ through ξ has codimension strictly greater than k and, therefore, ξ 6∈ Ω0 . The proof is complete. Proposition 3.22 and compactness of the Grassmannian imply that the mapping κ ˉ : Ω → Gn−k (Cn ) is meromorphic (by Definition 3.1). In Ω(k) it coincides with the holomorphic mapping κ, its graph is analytic, and sets κ ˉ (ξ) are compact for ξ ∈ Ω\Ω(k) . This fact allows to derive an upper estimate for the dimension of Ω sing . Theorem 3.23 The set Ωsing of essential singularities is an analytic subset of Ω with dim Ωsing ≤ n − 2. ˉ By Cartan– Proof. Let π denote the restriction of the projection (ξ, L) 7→ ξ to G. ˉ for which the dimension of the Remmert theorem (Theorem 3.6, (3)), points g ∈ G, ˉ which we denote by Σ. It level set π −1 (π(g)) at g is positive, form an analytic set in G, follows that Σ is analytic in Ω × Gn−k (Cn ). By the compactness of the Grassmannian ˉ to Ω. By Remmert Gn−k (Cn ) the projection π|Gˉ is a proper analytic mapping from G ˉ is analytic in Ω. However, proper mapping theorem (Theorem 3.8), the set π(G) ˉ π(G\G) = Ωsing , so that Ωsing is an analytic subset of Ω. The dimension estimate follows from [49, Satz 26]. Proposition 3.24 The mapping κ is holomorphically extendible over points ξ ∈ Ω\Ωsing . The set Ω\Ωsing is connected in Ω and hence κ has the holomorphic extension to ˉ to Ω\Ωsing . Moreover, for all Ω\Ωsing . This extension is equal to the restriction of κ sing sing ξ ∈ Ω\Ω and η ∈ (ξ + κ(ξ)) ∩ (Ω\Ω ) holds κ(ξ) = κ(η). Proof. By the definition of the set Ωsing , points ξ ∈ Ω\Ωsing are removable singularities of the mapping κ, which is holomorphically extendible to some neighborhood in Ω\Ωsing of each of these points. The set Ω\Ωsing is open and connected by Proposition 3.3, (3). Therefore, κ allows the global holomorphic extension to the whole of Ω\Ωsing , which we denote by κ0 . The graph of κ0 as a set in Ω × Gn−k (Cn ) lies between the graphs of κ and κ. ˉ Therefore, the closure of the graph of κ0 equals the graph of κ, ˉ so that κ0 is equal to the restriction of κ ˉ to Ω\Ωsing . Assume now that ξ and η satisfy conditions of Proposition 3.24. Let ξj ∈ Ω(k) , ξj → ξ. There exists a sequence ηj ∈ (ξj +κ(ξj )) ∩Ω(k) such that ηj → η. Hence κ(ηj ) → κ(η), because κ is extendible over η. But κ(ξj ) = κ(ηj ) because of ξj , ηj ∈ Ω(k) and condition (A3). We obtain κ(ξ) = limj κ(ξj ) = κ(η). Lemma 3.25 For every ξ ∈ Ωsing holds ξ + κ(ξ) ˉ ⊂ Ω\Ω(k) . Proof. Let ξ ∈ Ωsing and let C be an irreducible component of κ ˉ (ξ). Let C 0 be the e regular (smooth) part of C. Because Γ = Γ(ξ) is constant on ξ + κ(ξ), ˉ we get that for every η 0 ∈ C 0 with ξ + η 0 ∈ Ω holds Tη C 0 ⊂ ker DΓ(ξ + η 0 ). By Proposition 3.22, 34
(4), we have dim ker DΓ(ξ + η 0 ) ≥ n − k + 1, and hence ξ + η 0 ∈ Ω\Ω(k) . The set C 0 is dense in C and the set Ω is open. Therefore, every point η ∈ C is a limit of points η 0 ∈ C 0 such that ξ + η 0 ∈ Ω. Hence ξ + η ∈ Ω\Ω(k) . Since Ω\Ω(k) is closed in Ω, we e get ξ + C ⊂ Ω\Ω(k) . Because this is true for all irreducible components of κ(ξ), ˉ we e obtain ξ + κ(ξ) ˉ ⊂ Ω\Ω(k) . We will need the following two lemmas. We are interested in blowing-ups with non-smooth centers. If S is a smooth submanifold of Cn of codimension k, then the fibers of the blowing-up over the points of S are the projective spaces Pk−1 (cf. [35, VII.5], [12, II.4, IV.3, VI.2.2]). If S is an analytic set, then the described result is applicable to the interiors of the irreducible components of S of different dimensions, which leads to the projective spaces of possibly different dimensions at different points (cf. [31, II.7, II.8]). For us, it will be important what happens at singular points. Lemma 3.26 Let a mapping f : Ω → Cp be holomorphic in an open connected set Ω in Cn . Let p ≥ 2, f 6≡ 0, S = {ξ ∈ Ω : f (ξ) = 0}, ξ0 ∈ S, dimξ0 S ≤ n − 2. Then the mapping F : ξ 7→ Cf (ξ) : Ω\S → P(Cp ) does not allow a holomorphic extension over ξ. Condition dimξ0 S ≤ n − 2 means that the largest common divisor of functions f1 , . . . , fp does not vanish at ξ0 . Proof. Reasoning as in Lemma 3.14, one readily shows that if there is a hyperplane H in P(Cp ) such that Fˉ (ξ0 ) ∩ H is empty, then F allows the holomorphic extension to a neighborhood of ξ0 , which we also denote by F . Conversely, if F is holomorphic, then there exists H as above. In this case there exists an index i such that conditions η ∈ F (ξ0 ), η 6= 0 imply ηi 6= 0. Because F is continuous at ξ0 , there exists > 0 and a neighborhood U of the point ξ0 in Ω, such that conditions ξ ∈ U , η ∈ F (ξ) imply |ηi | ≥ |ηj | for all j 6= i. Therefore, if ξ ∈ U \S, then fi (ξ) 6= 0, or S ∩ U ⊂ fi−1 (0) ∩ U ⊂ S, which contradicts dim fi−1 (0) = n − 1 and dimξ0 S ≤ n − 2. This completes the proof. Let us give the following simple lemma from linear algebra without proof. Lemma 3.27 Let A ∈ Cp×n with rank A = k. Let the maximal rank of the matrix A k×k be attained on the submatrix AM with rows and columns with indices in L and L ∈C M, respectively, L ⊂ {1, . . . , p}, M ⊂ {1, . . . , n}. Then for every r, 1 ≤ r ≤ k, and {λi }ri=1 ⊂ L there exist {μi }ri=1 ⊂ M such that det Aμλii 6= 0, where Aμλii ∈ Cr×r is a submatrix of A obtained by the intersections of rows λi with columns μi , i = 1, . . . , r. To simplify notation let us denote ∂j Γi (ξ) by Aij (ξ). Note that the analysis is still valid in a more general case, when κ(ξ) is a fibration by the kernels of a holomorphic matrix valued mapping A : Ω → Cp×n . In this case, Gn−k (Cn ) 3 κ(ξ) = ker A(ξ) for ξ ∈ Ω(k) , rank A(ξ) ≤ k and Ω(k) 6= ∅. Proposition 3.28 Assume that Ωsing is not empty. Then dimξ Ωsing ≥ k − 1 for all ξ ∈ Ωsing .
35
Proof. In view of Lemma 3.27, there exist subsets L, M of {1, . . . , n} consisting of k elements such that the determinant ΔM L (ξ) = det Aij (ξ)i∈L,j∈M does not vanish identically in Ω. Let ZL,M denote the common zero set of functions ΔM L in Ω. Then for ξ ∈ Ω\ZL,M holds η ∈ ker A(ξ) if and only if X X Aij (ξ)ηj + Aim (ξ)ηm = 0, i ∈ L. (3.6) m6∈M
j∈M
Moreover, if we solve equations (3.6) in ηj , j ∈ M, equality (3.6) becomes equivalent to jm X fL,M (ξ) ηm , j ∈ M, ηj = (3.7) M ΔL (ξ) m6∈M
jm j (ξ) are polynomial in components of the matrix A(ξ). For j ∈ M, let ZL,M where fL,M jm denote the common zero set of functions fL,M , m 6∈ M, and ΔM L , after we divide them by their largest common divisor. Because Ω\ZL,M is dense in Ω and the right hand j side of (3.7) is ambiguous at ξ ∈ Ωsing , in view of Lemma 3.26 we have ZL,M ⊂ Ωsing . j j ≤ n − 2. The number of function in the definition of ZL,M is a Therefore, dimξ ZL,M sum of n − |M| = n − k indices m 6∈ M and an additional function ΔM . Hence L j ≥ n − (n − k + 1) = k − 1 dimξ ZL,M
and dimξ Ωsing ≥ k − 1. Remark 3.29 It follows from the proof of Proposition 3.28 that the statement is still valid in a more general case when we replace the Jacobian DΓ(ξ) by an arbitrary holomorphic matrix valued function A(ξ) ∈ Cp×n in Ω. In this case we still define k = maxξ∈Ω rank A(ξ) and κ(ξ) = ker A(ξ) ∈ Gn−k (Cn ) in Ω(k) .
Proposition 3.30 Let k = n − 1. If the set Ωsing is not empty, then dim Ωsing = n − 2. Moreover, dim A = n − 2 for every irreducible component A of the set Ωsing . Proof. The first statement follows from Proposition 3.28 and Theorem 3.23. Let L, M be as in the proof of Proposition 3.28. Then there exist indices l and m, 1 ≤ l, m ≤ n, jm k k k l 6∈ L, m 6∈ M. Denote Δlm = ΔM L , ZL,M = Zlm and fL,M = flm . It suffices to show that [ k Ωsing = Zlm . k 6≡0 l,m,k:Δlm 6≡0, flm
k It is clear from the proof of Proposition 3.28 that Zlm ⊂ Ωsing . Therefore, it is sufficient sing k to show that for every ξ ∈ Ω there exist indices l, m, k such that Δlm 6≡ 0, flm 6≡ 0 k and ξ ∈ Zlm . Using Lemma 3.27 with r = n − 1, we get that for every l there exists m such that Δlm 6≡ 0. Let us take ξ ∈ Ωsing . Then Δlm (ξ) = 0, because otherwise ξ ∈ Ω(n−1) . System (3.7) assumes the form
ηk =
k flm (ξ) ηm , k 6= m. Δlm (ξ)
k (ξ) ≡ 0, then all ηk ≡ 0, k 6= m, which would mean that If for all k ∈ M holds flm k the fibration is constant. Therefore, for every m there exist l and k such that flm 6≡ 0. k k Condition ξ 6∈ Zlm means that flm (ξ) 6= 0, or ηm = 0. If it were true for all m, we would get that all ηm = 0, which is impossible since Ωsing 6= ∅. The proof is complete.
36
Theorem 3.31 Let dim Ωsing = k − 1 and let A be the regular (smooth) component of Ωsing of dimension k − 1. Assume that there is a k-dimensional surface S containing A and transversal to Ω\Ω(k) . Let η ∈ Ω\Ωsing . Then the intersection of η + κ(η) and S can not be transversal at the points of A. In particular, for k = n − 1 holds κ(η) ⊂ Tξ A for all η ∈ κ(Ω ˉ sing )\Ωsing and ξ = (η + κ(η)) ∩ A. Proof. Suppose that there is η ∈ Ω\Ωsing such that the sets η + κ(η) and S intersect transversally. Let δ : Ω → Cn−k denote the mapping locally determining κ(ξ) at ξ ∈ Ω(k) . The mapping δ is meromorphic and ambiguous at the points of Ωsing . The limits of the fibers κ(ξ) for ξ → A are contained in the complement to Ω(k) in Ω (Lemma 3.25), and, therefore, are transversal to the surface S. It follows that all regular fibers κ(ξ) with ξ ∈ Ω(k) intersect S transversally, because dim S + dim κ e |Ω(k) (ξ) = k + (n − k) = n.
Hence the restriction δ|S is ambiguous at all points of the component A. This contradicts to the fact that the points of ambiguity of δ|S satisfy at least two independent analytic equations and hence form the set of codimension equal to at least 2 in S. This contradiction completes the proof of the first part of the theorem. For k = n − 1 we have dim A = n − 2 by Proposition 3.30. If the line η + κ(η) is not tangent to A, then there is a surface S, satisfying conditions of the first part of the theorem, which is impossible. The proof is complete. Let us prove a generalization of the second part of the Theorem 3.31 to other values of k. Proposition 3.32 Let A be a regular (smooth) component of Ωsing . Then for every e ξ ∈ A holds κ(ξ) ˉ ⊂ Tξ A.
e ˉ ⊂ Ωsing . Proof. The statement is obvious if for every ξ ∈ Ωsing we have ξ + κ(ξ) sing e e Suppose now that there is ξ ∈ A with ξ + κ(ξ) ˉ 6⊂ Ω and κ(ξ) ˉ 6⊂ Tξ A. The set κ ˉ (ξ) ˉ not contained in Ωsing and in is connected and hence there are different κ1 , κ2 ∈ κ(ξ), Tξ Ωsing (even locally). The set Ktr of all H ∈ Gk+1 (Cn ) such that H ∩ κ1 , H ∩ κ2 are not contained in Tξ Ωsing is open and dense in Gk+1 (Cn ). The set K0 = {H ∈ Gk+1 (Cn ) : H ∩ κ1 6= H ∩ κ2 } is open and dense in Gk+1 (Cn ). For i = 1, 2 the sets Ki = {H ∈ Gk+1 (Cn ) : dim H ∩ κi = 1} are open and dense in Gk+1 (Cn ). Therefore, the intersection of all these sets is also open and dense in Gk+1 (Cn ). Let H ∈ K1 ∩K2 ∩K0 ∩Ktr . Let η ∈ Ω(k) be close to ξ with κ(η) close to one of κi . In view of the transversality we have dim(η+κ(η))∩(ξ+H) = 1. The set Ω(k) ∩(η+κ(η)) is not empty and hence it is open and dense in Ω∩(η+κ(η)). It follows that there exists ζ ∈ Ω(k) ∩ (η + κ(η)), ζ sufficiently close to (η + κ(η)) ∩ (ξ + H) such that H0 ∈ K1 ∩ K2 ∩ K0 ∩ Ktr with ζ ∈ H0 . Without loss of generality we may assume that H = H0 . By (A3) we have κ(η) = κ(ζ), and for the mapping γ = Γ|(ξ+H)∩Ω 37
holds ker Dγ(ζ) = κ(ζ) ∩ H. This intersection is one dimensional and hence rank Dγ(ζ) = dim H − 1 = k. Moreover, if θ ∈ ker Dγ(ζ), then γ(θ) = Γ(θ) = Γ(ζ) = γ(ζ), because θ ∈ κ(ζ). Thus, conditions (A1), (A2) and (A3) are satisfied for the mapping γ. Let ηj ∈ Ω(k) be such that ηj → ξ and κ(ηj ) converges to one of κi . Because the set Ω(k) (H) = {ζ ∈ (ξ + H) ∩ Ω : rank Dγ(ζ) = k} is open and dense in (ξ + H) ∩ Ω, there is a sequence ζj ∈ Ω(k) (H) arbitrary close to (ξ + H) ∩ (ηj + κ(ηj )). Then the sequence κ(ζj ) is arbitrary close to κ(ηj ). We have Ω(k) (H) ⊂ Ω(k) and κ is constant on an open dense subset of (η + κ(η)) ∩ Ω, η ∈ Ω(k) . Therefore, at the point ξ in (ξ + H) ∩ (ξ + κ) we get all elements of H ∩ κ(ξ) ˉ as limits of sequences H ∩ κ(ζj ), ζj ∈ Ω(k) (H), ζj → ξ. In particular, H ∈ K0 implies that we get at least two different lines κi ∩ H ∈ κ(ξ) ˉ ∩ H in the limit. It means that ξ belongs to the set of essential singularities of the fibration ζ 7→ κ(ζ) ∩ H in H, corresponding to the mapping γ. Moreover, Tξ Ωsing (γ) ⊂ Tξ Ωsing (Γ) ∩ H and condition H ∈ Ktr implies H ∩ κi 6⊂ Tξ Ωsing (γ). This is a contradiction with Theorem 3.31 and Proposition 3.30, if we apply them to the mapping γ with n = k + 1. Now we can prove Theorem 3.9. Proof of Theorem 3.9. The second part follows from Proposition 3.32. From the e e ˉ inclusion κ(ξ) ˉ ⊂ Tξ Ωsing we get the estimate for the dimension dimξ Ωsing ≥ dim κ(ξ), which is ≥ n − k + 1 in view of Proposition 3.22. The second estimate follows from Proposition 3.28.
3.3
Further results
Let us briefly discuss some further results related to the fibrations with affine fibers. It turns out that an important tool in the analysis of the set Ωsing is the equivalence relation induced by the fibration κ. Two points of the set Ω(k) are called equivalent if they belong to the same fiber. The closure of such equivalence relation defines a relation in Ω, for which the equivalence class of a point ξ ∈ Ω consists of the limit cone e κ(ξ). ˉ In particular, for points ξ ∈ Ω0 = Ω\Ωsing the equivalence classes have the same dimension as the classes of points ξ ∈ Ω(k) . Proposition 3.33 The set
R = {(ξ, η) ∈ Ω0 × Ω0 : η − ξ ∈ κ(ξ)} is a closed analytic subset of Ω0 × Ω0 of dimension 2n − k. It defines an equivalence relation on Ω0 . The equivalence class of a point ξ ∈ Ω0 is R(ξ) = {η ∈ Ω0 : (ξ, η) ∈ R}. Therefore, the set Ω0 is the union of the equivalence classes R(ξ). The factor space Ω0 /R is a smooth analytic manifold of dimension k.
38
Let ρ denote the natural projection ρ : Ω0 3 ξ 7→ R(ξ) ∈ Ω0 /R. The factorization Ω0 /R is well defined since R is closed in Ω0 × Ω0 . The factor space Ω0 /R is equipped with the strongest Hausdorff topology, for which the projection ρ is continuous. The mapping Γ is constant on the equivalence class R(ξ). It follows that there exists an analytic mapping g : Ω0 /R → Cp such that Γ = g ◦ ρ.
(3.8)
Because the condition rank DΓ = k implies rank Dg ≥ k, the tangent mapping to g is everywhere injective and the mapping g is an immersion from Ω(k) /R to Cp . Factorization (3.8) implies the holomorphic factorization condition for the wave fronts (1.8). Let π1 and π2 be the natural holomorphic projections π1 : (ξ, η) 7→ ξ : Ω × Ω → Ω, π2 : (ξ, η) 7→ η : Ω × Ω → Ω. The set R ∩ (Ω(k) × Ω(k) ) is the complement to the analytic set ((Ω\Ω(k) ) × Ω) ∪ (Ω × (Ω\Ω(k) ))
(3.9)
in the analytic set {(ξ, η) ∈ Ω × Ω : η − ξ ∈ ker DΓ(ξ)}
ˉ in Ω × Ω is an in Ω × Ω. Therefore, R is analytically constructible and its closure R e analytic subset of Ω × Ω (see Theorem 3.7). The cone κ(ξ) ˉ over κ(ξ) ˉ can be written ˉ By Theorem 3.23 the set Ωsing is analytic in Ω. Hence the set as π2 (π1−1 ({ξ}) ∩ R). ˉ ∩ (Ωsing × Ω) → Ωsing ˉ ∩ (Ωsing × Ω) is also analytic in Ω × Ω and projection π1 : R R is surjective. For the second projection we have [ ˉ ∩ (Ωsing × Ω)). e (ξ + κ(ξ)) ˉ = π2 (R (3.10) ξ∈Ωsing
ˉ ∩ (Ω0 × Ω0 ) is analytically constructible and its Proposition 3.34 The set R = R ˉ closure R is an analytic subset of Ω × Ω for which holds ˉ dim(R\R) ≤ 2n − k − 1. Moreover,
ˉ ∩ (Ωsing × Ω)) ≤ n − 1, n − k + 1 ≤ dim π2 (R
(3.11)
× Ω)).
(3.12)
dim Ω
sing
ˉ ∩ (Ω + n − k + 1 ≤ dim(R
sing
Proof. The first estimate follows from the fact that the set R ∩ (Ω(k) × Ω(k) ) is dense ˉ and hence in R ˉ ˉ = 2n − k. dim(R\(R ∩ (Ω(k) × Ω(k) ))) < dim R 39
Estimate (3.11) follows from (3.10), Lemma 3.25 and inequality dim Ω\Ω(k) ≤ n − 1. ˉ ∩ (Ωsing × Ω)) has dimension ≥ n − k + 1 by For w ∈ Ωsing the level set π1−1 (w) ∩ (R Proposition 3.22. By Theorem 3.4 we get (3.12). Note that estimate dim Ωsing ≤ n−2 of Theorem 3.23 follows from Proposition 3.34, ˉ ∩ (Ωsing × Ω) is contained in R\R ˉ (1). Indeed, R and hence its dimension ≤ 2n − k − 1. sing ˉ The level sets of π1 from R∩(Ω ×Ω) to Ωsing have dimension ≥ n−k +1. Therefore, dim Ωsing ≤ n − 2 follows from Theorem 3.4. In the sequel we will assume that the set Ωsing is not empty. The following statement is given without proof. Proposition 3.35
ˉ ∩ (Ωsing × Ω)) ⊂ Ω\Ω(k) hold. (1) Inclusions Ωsing ⊂ π2 (R
ˉ ∩ (Ωsing × Ω)), then the closure R ˉ is the union of equivalence (2) If Ωsing = π2 (R sing sing sing relations in (Ω\Ω ) × (Ω\Ω ) and in Ω × Ωsing . In other words, ˉ ∩ (Ωsing × Ω) ⊂ Ωsing × Ωsing . R Moreover, for all ξ ∈ Ωsing .
n − k + 1 ≤ dimξ Ωsing
ˉ ∩ (Ωsing × Ω)) and (3) If Ωsing 6= π2 (R l=
max
sing ×Ω\Ωsing )) ˉ η∈π2 (R∩(Ω
dim(Ωsing ∩ (η + κ(η))),
then dim π1 (R ∩ (Ωsing × Ω\Ωsing )) ≤ k − 2 + l.
(3.13)
ˉ ∩ (Ωsing × Ω\Ωsing )) = dim(R ˉ ∩ (Ωsing × Ω)) (4) If in addition to (3) we have dim(R sing sing sing ˉ ∩ (Ω or dim π1 (R × Ω\Ω )) = dim Ω , then ˉ ∩ (Ωsing × Ω)). dim Ωsing + 2 ≤ dim π2 (R
(3.14)
ˉ ∩ (Ωsing × Ω)). Then for all ξ ∈ Ωsing with ξ 6∈ π1 (R ˉ ∩ (Ωsing × (5) Let Ωsing 6= π2 (R sing sing e Ω\Ω )) holds κ(ξ) ˉ ⊂ Ω . Moreover, in this case we also have k ≥ 3 and dimξ Ωsing ≥ n − k + 1. As a consequence we get the following estimates.
Proposition 3.36
ˉ ∩ (Ωsing × Ω\Ωsing ) ⊂ R ˉ ∩ (Ωsing × Ω) is proper. (1) The subset R
ˉ ∩ (Ωsing × Ω\Ωsing )) 6= Ωsing , then dimξ Ωsing ≥ n − k + 1 for all ξ ∈ (2) If π1 (R sing ˉ ∩ (Ωsing × Ω\Ωsing )) and k ≥ 3. Ω \π1 (R
ˉ ∩ (Ωsing × Ω\Ωsing )) = Ωsing , then dim Ωsing ≤ n − 3. (3) If π1 (R
40
4 4.1
Affine fibrations of gradient type Localization
In this section we continue to always assume that Γ satisfies conditions (A1)–(A3) of Introduction. Without loss of generality we can assume that the origin of the space Cn belongs to the open set Ω. Clearly, if the fibration κ is continuous at zero, that is, if 0 6∈ Ωsing , then there is a neighborhood U of the origin, disjoint from Ωsing , such that the fibers through the points of U are close to the fiber κ(0) through zero. Therefore, there exists a k–dimensional linear subspace of Cn , transversal to all κ(ξ) for ξ ∈ U . On the other hand, if 0 ∈ Ωsing and U ⊂ Ω is a small open neighborhood of zero, then for every H ∈ Gk (Cn ), either the set UH = {ξ ∈ U ∩ Ω(k) : κ(ξ) ∩ H = {0}}
is empty, or it is open and dense in U ∩ Ω(k) , and hence in U . Thus, for H with non-empty set UH , we can choose a coordinate system around zero such that H is parameterized by points (h, 0) ∈ Ck × Cn−k and that on an open dense subset of the set Ω(k) all fibers are transversal to the k–dimensional subspace of (h, 0) ∈ Ck × Cn−k . (k) Let πh : Cn → Ck denote the projection to the first k coordinates. Let Ωh be the set of all points h ∈ πh (Ω) with (h, 0) ∈ Ω(k) . The set of all h ∈ πh (Ω) such that (h, 0) ∈ Ω we denote by Ωh . For the simplicity in notations throughout this section we will identify Cn with 1×n , so that the elements of Cn are the rows with n elements in C. In this notation C (k) we parameterize the fibration by a matrix valued mapping R : Ωh ⊂ Ck → C(n−k)×k . The fibers κ(h, 0) can be parameterized by λ in some neighborhood of the origin in Cn−k : locally we have κ(h, 0) = {(h + λR(h), λ), λ ∈ Cn−k } = {(h +
n−k X i=1
λi Ri (h), λ), λ ∈ Cn−k }.
The rows of R(h) will be denoted by Ri (h) ∈ C , i = 1, . . . , n − k, and the components of each row Ri (h) by Rij (h), j = 1, . . . , k. The j-th column of R(h) we denote by Rj (h) ∈ Cn−k , j = 1, . . . , k, and its elements by Rij (h). In order to avoid any confusion with this notation, we write k z }| { R11 R12 ... R1k . . . .. .. . R = (n − k), . ∙∙∙ 1 2 k Rn−k Rn−k . . . Rn−k (4.1) 1 2 k = R l Rl . . . R l , Rl ∂1 Rl1 . . . ∂k Rl1 .. DRl = ... . ∙∙∙ . k
∂1 Rlk
...
∂k Rlk
Condition (A3) that Γ is constant on κ can now be written as (k)
Γ(h + λR(h), λ) = Γ(h, 0), ∀h ∈ U ∩ Ωh , λ ∈ V, 41
(4.2)
in some neighborhood U × V ⊂ (Ck × Cn−k ) ∩ Ω of the origin. Lemma 4.1 Define γ : Ωh → Cp by γ(h) = Γ(h, 0). Then " # n−k X k {∂i Γ(h + λR(h), λ)}i=1 Ik + λl DRl (h) = Dγ(h).
(4.3)
l=1
(k)
for all h ∈ Ωh . Proof. Differentiating Γj in (4.2) with respect to hm , we obtain ∂m Γj (h, 0)
= = +
∂ ∂hm Γj (h
+ λR(h), λ) m ∂ m Γj (h + λR(h), λ)[1 + λ∂m R (h)] P i 1≤i≤n−k,i6=m ∂i Γj (h + λR(h), λ)[λ∂m R (h)].
In matrix notation it means (4.3). The group GLn (C) acts on Γ in a natural way. For A ∈ GLn let us denote ΓA (ξ) = Γ(Aξ). Proposition 4.2 Let Γ satisfy conditions (A1)–(A3). (1) For every A ∈ GLn the mapping ΓA : A−1 (Ω) → Cp satisfies (A1)–(A3). Moreover, Ω(k) (ΓA ) = A−1 (Ω(k) (Γ)) and Ωsing (ΓA ) = A−1 (Ωsing (Γ)). (2) There exists A ∈ GLn , such that the mapping γA : (A−1 (Ω))h → Cp defined by γA (h) = ΓA (h, 0) satisfies rank DγA (h) = k for h in an open dense subset of (A−1 (Ω))h . Proof. Statement (1) is straightforward. Now we will prove (2). Let i1 , . . . , ik be the minimal indices, for which there exist j1 , . . . , jk and ξ ∈ Ω(k) , such that ∂i1 Γj1 (ξ) . . . ∂ik Γj1 (ξ) .. .. det (4.4) 6= 0. . ∙∙∙ . ∂i1 Γjk (ξ)
...
∂ik Γjk (ξ)
For τ 6= 0 define Aτ ∈ GLn by
Aτ ζ = τ ζ +
k X
ζl eil ,
l=1
where el stands for the l-th standard unit basis vector of Cn . By the first part of the proposition the mapping ΓAτ satisfies conditions (A1)-(A3). Jacobian of the mapping γAτ has the form τ ∂1 Γ1 (Aτ ζ) + ∂i1 Γ1 (Aτ ζ) . . . τ ∂k Γ1 (Aτ ζ) + ∂ik Γ1 (Aτ ζ) .. .. .. DγAτ (ζ) = . (4.5) . . . τ ∂1 Γp (Aτ ζ) + ∂i1 Γp (Aτ ζ)
42
...
τ ∂k Γp (Aτ ζ) + ∂ik Γp (Aτ ζ)
In particular, this matrix contains a block of the form τ ∂1 Γj1 (Aτ ζ) + ∂i1 Γj1 (Aτ ζ) . . . τ ∂k Γj1 (Aτ ζ) + ∂ik Γj1 (Aτ ζ) .. .. .. . . . . τ ∂1 Γjk (Aτ ζ) + ∂i1 Γjk (Aτ ζ) . . . τ ∂k Γjk (Aτ ζ) + ∂ik Γjk (Aτ ζ)
(4.6)
(k) The determinant of this block does not vanish for some ζ ∈ A−1 ) in view of the τ (Ω choice of i1 , . . . , ik and j1 , . . . , jk as in (4.4), if τ is sufficiently small. Finally, in view of the analyticity of Γ, the block (4.6) is non-degenerate on the complement of an analytic set in Ω(k) h , implying the last statement. In view of Proposition 4.2 without loss of generality we may assume that the mapping γ in (4.3) satisfies the non-degeneracy condition of Proposition 4.2, (2). Thus, for the global fibrations we get
Proposition 4.3 Let Γ : Cn → Cp be an entire holomorphic function. Then for every (k) h ∈ Ωh the condition rank Dγ(h) = k implies that the matrices DRl (h) are nilpotent for all 1 ≤ l ≤ n − k.
Proof. Suppose that some DRl0 (h) is not nilpotent and let μ ∈ C be its nonzero eigenvalue. Equation (4.3) holds locally in h, λ, but both sides are holomorphic with respect to λ, so by the analytic continuation it holds for all λ ∈ C, because Γ is an entire function. Substitution of λl = −μ−1 δll0 leads to the degeneracy of Dγ(h), a contradiction. Example 4.4 Define Γ : C4 → C4 by Γ(ξ1 , ξ2 , ξ3 , ξ4 ) = (ξ1 ξ2 + ξ3 ξ4 , ξ2 , ξ3 , 0). The fibration associated to Γ is global, and for the mapping Γ and its restriction γ(h1 , h2 , h3 ) = (h1 h2 , h2 , h3 , 0) holds
ξ2 0 DΓ(ξ) = 0 0
ξ1 1 0 0
ξ4 0 1 0
ξ3 0 , Dγ(h) = 0 0
h2 0 0 0
0 0 . 1 0
h1 1 0 0
h2 6= 0. Localization R of the fibration Proposition 4.3 holds for all h ∈ C3 with κ satisfies the equality DΓ(h, 0) R(h) = 0, from which we conclude that R1 (h) = 1 0 h3 h−2 −h−1 2 2 2 3 −h3 /h2 , R (h) = 0, R (h) = 0, with nilpotent matrix DR(h) = 0 0 . 0 0
0
0
Example 4.5 The rank of the matrix DΓ can drop by a number larger than one at points of Ω\Ω(k) . For Γ1 (ξ) = (ξ1 ξ2 + ξ3 ξ4 , ξ22 , ξ32 , 0) and Γ2 (ξ) = 12 ((ξ1 ξ2 + ξ3 ξ4 )2 , ξ22 , ξ32 , 0) we have ξ4 ξ3 ξ2 ξ1 ξ2 α(ξ) ξ1 α(ξ) ξ4 α(ξ) ξ3 α(ξ) 0 2ξ2 0 0 0 ξ2 0 0 , DΓ2 (ξ) = DΓ1 (ξ) = 0 0 2ξ3 0 0 0 ξ3 0 0 0 0 0 0 0 0 0 with α(ξ) = ξ1 ξ2 + ξ3 ξ4 . Therefore, rank DΓ1 |ξ2 =ξ3 =0 = 1, rank DΓ1 |ξ=0 = 0 and rank DΓ2 |ξ2 =ξ3 =0 = 0. 43
4.2
Fibrations of gradient type
In what follows we prove the results simultaneously for two fields of scalars, R and C, which we will denote by K. In this notation for m ≥ 1 we will write C m (Kn ) for the space C m (Rn ) when K = R, and for the space of holomorphic functions when K = C. We consider the case of the fibration by codimension k hyperplanes, corresponding to the conditions (A1)–(A3) with Γ having a gradient form, that is Γ(ξ) = ∇ψ(ξ)
(4.7)
for some function ψ : Ω → K. If the mapping Γ is as in (4.7), the associated fibration will be called the fibration of gradient type. Fibrations of this type will appear in Section 5, where Γ is of the form (4.7) with a generating function φ or its non-homogeneous version ψ. In this case DΓ = D2 ψ and fibers κ of the mapping Γ correspond to the level sets of the gradient ∇ψ. In general, we will make some smoothness assumptions, but we note, that the following statements are valid where all the derivatives make sense, so at least at the open subset Ω(k) with rank D2 ψ = k. We adopt notation (4.1), related to the mapping (k) R : Ωh ⊂ Kk → K(n−k)×k as in the previous section for the case Γ = ∇ψ as well.
Theorem 4.6 (1) Let ψ ∈ C 2 (Kn ). The fibration by the level sets of ∇ψ is given by the mapping R : Kk → K(n−k)×k in a neighborhood U × V ⊂ Kk × Kn−k of the origin, i.e. (k)
if and only if
∇ψ(h + λR(h), λ) = ∇ψ(h, 0), ∀h ∈ U ∩ Ωh , λ ∈ V,
2
D ψ(h + λR(h), λ)
RT (h) In−k
(4.8)
(k)
= 0 ∈ Kn×(n−k) , ∀h ∈ U ∩ Ωh , λ ∈ V.
(4.9)
In this case ψ must be of the form (4.10) with φ(h) = ψ(h, 0), χk+l (h) =
∂ψ (h, 0), l = 1, . . . , n − k. ∂hk+l (k)
If ψ ∈ C 3 (Kn ), then φ and χk+l satisfy equations (4.11),(4.12) in Ωh . (2) Let function ψ be defined by
ψ(h + λR(h), λ)
= =
T R (h) φ(h) + hλ, ∇ψ(h, 0) i In−k Pn−k Pk φ(h) + l=1 λl [ j=1 ∂j φ(h)Rlj (h) + χk+l (h)],
where φ ∈ C 3 (Kn ) solves the system of equations k X i=1
k
1 2 (k
(4.10)
− 1)k(n − k) partial differential
X ∂ 2 φ ∂Ri ∂ 2 φ ∂Rli l − = 0, 1 ≤ m < j ≤ k, 1 ≤ l ≤ n − k (4.11) ∂hi ∂hj ∂hm i=1 ∂hi ∂hm ∂hj 44
and let functions χk+l ∈ C 2 (Kn ), l = 1, . . . , n − k, satisfy ∂j χk+l (h) + 1 ≤ j ≤ k.
Pk
i=1
∂j ∂i φ(h)Rli (h) = 0.
(4.12)
Then ψ satisfies the factorization condition (4.8) locally in h, λ, for which ! n−k X det Ik + λl DRl (h) 6= 0, (4.13) l=1
Proof. (1) Differentiation of (4.8) with respect to λ yields ∂ ∂λi ∂j ψ(h
+ λR(h), λ)
= + =
Pk
∂m ∂j ψ(h + λR(h), λ)Rim (h) ∂k+i ∂j ψ(h + λR(h), λ) T Ri (h) ∇∂j ψ(h + λR(h), λ) . δj,k+i m=1
(4.14)
In view of (4.8) this is equal to zero. Because this holds for all 1 ≤ j ≤ n and 1 ≤ i ≤ n−k, we get (4.9). Now assume (4.9) to hold. Differentiating ∂j ψ(h+λR(h), λ) for 1 ≤ j ≤ n with respect to λi , we obtain equation (4.14). Writing this for all i, j in matrix notation, we get T R (h) 2 ∇λ ∇ψ(h + λR(h), λ) = D ψ(h + λR(h), λ) , In−k the latter equals zero by (4.9). Hence ∇ψ(h + λR(h), λ) = const(h) = ∇ψ(h, 0), which is (4.8). Differentiating ψ(h + λR(h), λ) with respect to λ, we obtain T T R (h) R (h) = ∇ψ(h, 0) , ∇λ ψ(h + λR(h), λ) = ∇ψ(h + λR(h), λ) In−k In−k where the last equality follows from (4.8). Thus, ψ must have the form (4.10). Now let us prove (4.11). Writing (4.9) in a scalar form once again, and substituting λ = 0, the equalities for 1 ≤ j ≤ k yield k X i=1
∂i ∂j φ(h)Rli (h) + ∂k+l ∂j ψ(h, 0) = 0, 1 ≤ l ≤ n − k,
which are equations (4.12) in view of ψ ∈ C 2 (Kn ) and ∂j ∂k+l ψ = ∂k+l ∂j ψ. The relations ∂m ∂j χk+l = ∂j ∂m χk+l for ψ ∈ C 3 (Kn ) and (4.12) imply k k ∂ X ∂ X ( ∂i ∂j φ(h)Rli (h)) = ( ∂i ∂m φ(h)Rli (h)). ∂hm i=1 ∂hj i=1
In view of φ ∈ C 3 these equations imply (4.11). 45
(2) Define ψ as in (4.10). First we observe that equations (4.11) guarantee the existence of χk+l ∈ C 2 (Kn ) satisfying (4.12). Differentiating (4.10) with respect to hm , 1 ≤ m ≤ k, we get P ∂m ψ(h + λR(h), λ)(1 + λ∂m Rm (h)) + 1≤i≤k,i6=m ∂i ψ(h + λR(h), λ)(λ∂m Ri (h)) = ∂h∂m ψ(h + λR(h), λ) Pn−k Pk = ∂m φ(h) + l=1 λl [ j=1 (∂m ∂j φ(h)Rlj (h) + ∂j φ(h)∂m Rlj (h)) + ∂m χk+l (h)] Pn−k Pk = ∂m φ(h) + l=1 λl [ j=1 ∂j φ(h)∂m Rlj (h)],
where we used definitions (4.10) and (4.12). In matrix notation we get ∇k1 ψ(h + λR(h), λ)(Ik +
n−k X l=1
λl DRl (h)) = ∇k1 ψ(h, 0)(Ik +
n−k X
λl DRl (h))
l=1
and for h, λ satisfying (4.13) we get ∂j ψ(h + λR(h), λ) = ∂j ψ(h, 0), 1 ≤ j ≤ k, independent of λ. For the other components of ∇ψ we differentiate (4.10) with respect to λl and get Pk j j=1 ∂j ψ(h + λR(h), λ)Rl (h) + ∂k+l ψ(h + λR(h), λ) ∂ = ∂λ ψ(h + λR(h), λ) Pkl j = j=1 ∂j φ(h)Rl (h) + χk+l (h), the last equation follows from definition (4.10). This implies ∂k+l ψ(h + λR(h), λ) = χk+l (h) in view of the just proved factorization property for 1 ≤ j ≤ k. The proof is complete. It follows from Theorem 4.6 that being a fibration of gradient type is equivalent to the system of equations (4.11) if we regard (4.12) as a definition of χk+l . Expression (4.11) in the matrix form becomes ∂m Rl (h)(D2 φ(h))j = ∂j Rl (h)(D2 φ(h))m , 1 ≤ l ≤ n − k,
for each row Rl of R, where (D2 φ(h))j is the j-th column of D2 φ(h). Denoting by k×k Ksymm the space of symmetric k × k-matrices, we conclude Corollary 4.7 Condition (4.11) is equivalent to the condition ∂1 Rl (h) 2 (k) .. k×k DRl (h)T D2 φ(h) = D φ(h) ∈ Ksymm , ∀h ∈ Ωh , 1 ≤ l ≤ n − k. . ∂k Rl (h)
Note, that in our notation ∂m Rl (h) is the m-th column of the matrix DRl (h), and the m-th row of DRl (h)T , which is the reason for the transpose. Remark 4.8 Suppose that instead of the gradient form (4.7) we assume that both DΓ and Dγ are symmetric square matrices after a possible elimination or addition of dependent rows. Then the symmetricity condition of Corollary 4.7 P follows from equation (4.3) of Lemma 4.1, after multiplication from the left by (Ik + l λl DRl (h))T and using the symmetricity of DΓ and Dγ. However, the matrix DR(h)T Dγ(h) in the examples of Section 4.1 need not be symmetric, in comparison with Corollary 4.7. 46
4.3
Reconstruction of the phase function
The construction of the function ψ in Theorem 4.6, (2), clearly depends on the singularities of the fibration R. In the case of K = R by C m (U ) we denote the usual space of m times continuously differentiable functions in U , 1 ≤ m ≤ ∞, and analytic functions with m = ω. In the complex case K = C all these spaces coincide with the space of holomorphic functions in U . Let Ω0 be the set of all points in Ω satisfying condition (4.13): Ω0 = {(h, λ) ∈ Ω ∩ (Kk × Kn−k ) : det(Ik +
n−k X l=1
λl DRl (h)) 6= 0}.
(4.15)
Assume now that a fibration R is of class C m in an open subset Ωa of Ωh , φ is in C m+1 (Ω) and all χk+l ∈ C m (Ω). Then the mapping Φ : Ω ∩ (Kk × Kn−k ) 3 (h, λ) 7→ (Φ1 (h, λ), Φ2 (h, λ)) ∈ Kk × Kn−k with components Φ1 (h, λ) Φ2 (h, λ)
= =
h+ λ
is C m on πh−1 (Ωa ) ∩ Ω and its Jacobian is det DΦ(h, λ) = det
Ik + λDR(h) 0
Pn−k l=1
R(h) In−k
λl Rl (h),
(4.16)
= det(Ik +
n−k X
λl DRl (h)),
l=1
which is not zero when (h, λ) ∈ Ω0 . By the implicit function theorem the mapping Φ−1 is also C m in Φ(πh−1 (Ωa ) ∩ Ω0 ). The statement of Theorem 4.6, (2), implies that on Ω0 the function ψ satisfies (ψ ◦ Φ)(h, λ) = φ(h) +
k X j=1
∂j φ(h)(Φ1 (h, λ) − h)j +
n−k X
λl χk+l (h),
l=1
implying that ψ ∈ C m in Φ(πh−1 (Ωa ) ∩ Ω0 ) ⊂ Φ(Ω). Thus, we arrive at Proposition 4.9 Let R : Ωa ⊂ Ωh ⊂ Kk → K(n−k)×k and let Ωa be an open subset of Ωh . Let 3 ≤ m ≤ ∞ or m = ω, R ∈ C m (Ωa ), φ ∈ C m+1 (Ωh ), χk+l ∈ C m (Ωh ). Let Ω0 be given by (4.15) and let Φ be as in (4.16). Then the function ψ defined in (4.10) is of class C m in Φ(πh−1 (Ωa ) ∩ Ω0 ) ⊂ Φ(Ω). Assume now, the the sets πh−1 (Ωa ), Ω and Ω0 are locally equal, let U denote their open intersection. In particular, this is the case with m = ω, when the fibration is locally analytic, given by an analytic mapping R, for which the condition (4.13) is locally satisfied. As it is shown in Proposition 4.3, it can hold even globally if all DRl are nilpotent. According to Theorem 4.6, the system of partial differential equations (4.11) is a sufficient condition for the existence of a function ψ, for which the fibration by the level sets of ∇ψ is locally given by R. Proposition 4.9 implies that ψ is analytic in Φ(U ).
47
However, the existence of a solution φ in C m of system (4.11) of second order partial differential equations is not automatic. In the complex case (K = C) one can use Cauchy–Kovalevskaya theorem, to obtain an analytic solution φ in a neighborhood of a point where not all coefficients of the iterated derivatives of the highest order vanish. In this case we obtain an analytic solution even for the Cauchy problem, prescribing φ on some non-characteristic hypersurface. It would be interesting to obtain generalizations of Proposition 4.9. In general, given a fibration defined by a mapping R ∈ C m , is it possible to construct a function ψ, for which the fibration by the level sets of ∇ψ is locally given by R? What are the smoothness properties of such ψ if it exists and when can we guarantee its smoothness? A special case occurs if the fibration R is analytic. According to the remarks above on Cauchy–Kovalevskaya theorem, it may imply the analyticity of φ and χ. But ψ, constructed by the formulae of Theorem 4.6, can be complex valued, even restricted to the real domain. Another interesting question, corresponding to the holomorphically extended real valued analytic functions is, can ψ be chosen in such a way, that restricted to Rn it defines a real valued analytic function. Or whether there exists ψ such that the imaginary part of ψ has a prescribed constant sign. In the latter case one can still regard ψ as an admissible phase function (see Section 5.3 and Remark 5.2). In general, the approach suggested by Theorem 4.6 reduces the questions posed above to the analysis of a large system of second order partial differential equations (4.11). The properties of its solutions depend on the type of the system, which is determined by the sign of its determinant, which in turn is an expression in terms of the fibration R. In general, if the fibration is sufficiently smooth, then the coefficients of system (4.11) are sufficiently smooth as well. In this case some properties can be found in [8], [5]. For hyperbolic equations cf. [6]. For fibrations with essential singularities the coefficients of system (4.11) are meromorphic and are given by rational functions. In the case of a three dimensional space with n = 3 the system consists of only one equation. It simplifies the analysis. Assume now that n = 3 and k = 2. By Theorem 3.10 for the case K = C fibrations with affine fibers are regular (analytic) everywhere. Hence we will assume smoothness in the real case K = R as well. System (4.11) for φ : K2 → K has now the form ∂ 2 φ ∂R1 ∂2φ ∂R1 ∂ 2 φ ∂R2 ∂R2 + − − = 0. (4.17) ∂h21 ∂h2 ∂h1 ∂h2 ∂h2 ∂h22 ∂h1 ∂h1 Proposition 4.10 (1) If the matrix DR(h) has no real eigenvalues, then equation (4.17) is elliptic. In this case the fibration is extendible globally in λ ∈ R, locally uniformly in h. (2) If the matrix DR(h) has a real non-zero eigenvalue, then the determinant of the matrix D2 φ(h) vanishes. (3) Equation (4.17) is elliptic, parabolic, hyperbolic, if the eigenvalues of the matrix DR(h) are not real, real equal to each other, real and distinct, respectively. (4) Let ψ ∈ C ω (R3 ), h ∈ R2 . If the determinant of the matrix D2 φ(h) vanishes, then there exists a neighborhood U of the point h such that equation (4.17) is parabolic in U . The same conclusion still holds if ψ ∈ C 3 (R3 ) and the fibration is globally extendible in λ, locally uniformly in h. 48
Proof. (1) If DR(h) has no real eigenvalues, then (4.13) is satisfied for all real λ, which implies that the fibration extends globally inλ. We denote Rij (h)= ∂j Ri (h). R12 R11 − μ = 0, the The equation for the eigenvalues of DR(h) is det R21 R22 − μ discriminant of which is d(h) = (R11 (h) + R22 (h))2 − 4 det DR(h) and our assumption means d(h) < 0. Now, the type of the equation (4.17) is defined by the sign of 1 R12 [R22 − R11 ] 2 det , 1 −R21 2 [R22 − R11 ] which is equal to −R12 R21 −1/4(R22 −R11 )2 = −1/4(R22 +R11 )2 +det DR = −d(h)/4. This means that the equation (4.17) is elliptic. (2) Suppose that μ ∈ R, μ 6= 0, is an eigenvalue of DR(h): DR(h)v = μv. Differentiating (4.8) with respect to h, we get ∂12 ψ(h + λR(h), λ) ∂1 ∂2 ψ(h + λR(h), λ) (I + λDR(h)) = D2 φ(h). ∂2 ∂1 ψ(h + λR(h), λ) ∂22 ψ(h + λR(h), λ) Now we apply both sides to v and take λ = −μ−1 to conclude that D2 φ(h)v = 0. (3) Follows from part (1). (4) The continuity of D2 φ(h) implies that there exists a neighborhood U of h, such that D2 φ is nonsingular in U . The argument of Proposition 4.3 implies that DR is nilpotent in U , its both eigenvalues are equal to zero. The second part of the proposition implies that (4.17) is parabolic. This completes the proof of Proposition 4.10. In the complex case (K = C) with a holomorphic fibration R, if for i = 1 or i = 2 we denote by hi the variable such that ∂Ri /∂hj , j 6= i, in (4.17) is locally non-zero at hi = 0, then by Cauchy–Kovalevskaya theorem equation (4.17) with holomorphic Cauchy boundary conditions ∂hl i φ, l < 2, is locally soluble and the solution is unique and holomorphic. In general, if the Cauchy boundary conditions are holomorphic on some non-characteristic for (4.17) hypersurface S in C2 , then there exists a unique solution φ of (4.17) locally at every point of S ([33, Vol.I, Theorems IX.4.5-IX.4.7]). In the real case (K = R) the smoothness properties of solutions of equation (4.17) depend on its type. Thus, if it is elliptic with smooth (real analytic) coefficients, it always has smooth (real analytic) solutions: Theorem 4.11 Let R : U ⊂ R2 → R2 be smooth (real analytic). Assume that for every h ∈ U the matrix DR(h) has no real eigenvalues. Then there exists a smooth (real analytic) function ψ : Ω ⊂ R3 → R, such that the fibration defined by the level sets of ∇ψ is given by R in U . Proof. According to Proposition 4.10 equation (4.17) is elliptic. This implies that its solutions ψ are smooth (analytic) if the operator in (4.17) has smooth (analytic) coefficients, see Corollary 8.3.2 for W F (φ) and Theorem 9.5.1 for W FA (φ) in [33], respectively. Then also all χk+l are smooth (analytic) and Proposition 4.9 implies the statement. Let us discuss briefly the case when equation (4.17) is strictly hyperbolic. If a mapping (fibration) R is prescribed on a smooth hypersurface S which is non-characteristic for (4.17), then solution φ is in the Sobolev space H (s) if the Cauchy data are in the 49
Sobolev spaces H (s) . By the Sobolev embedding theorem applied to all s > 0 we get that solution φ is smooth if the Cauchy data are smooth on S. For real analytic fibrations R let us use the methods described in [21, VI]. Let Ω be a connected complex analytic manifold of some dimension p. Let D(Ω) be the algebra of linear partial differential operators with holomorphic coefficients in Ω. Let J be a left ideal in D(Ω). Then J is called holonomic in Ω if for every x ∈ Ω the common zero set in Cp of the principal symbols q x (θ) at x of the Q ∈ J (in local coordinates) is equal to zero. In particular, the holonomicity assumption for the ideal generated by a hyperbolic operator P leads to an ordinary differential equation along every curve in the complement of the complex bicharacteristic cone of the principal symbol of P and the standard analytic continuation for ordinary differential equations can be applied. Let us apply [21, VI-16, Theorem] to our situation without giving many details. Proposition 4.12 Assume that a mapping R is analytic in an open subset Ω0 of R2 and that DR has real distinct eigenvalues in Ω0 . Let Ω be the complexification of Ω0 and let φ be a solution of (4.17). Assume also that the ideal J = {Q ∈ D(Ω) : Qφ = 0} generated by equation (4.17) is holonomic in Ω. Then φ is real analytic in Ω0 and has a holomorphic extension φγ along each curve γ in Ω starting in Ω0 . The holomorphic extension φγ also satisfies equations Qφγ = 0 for all Q ∈ J. Let us give several corollaries of Theorem 4.6 for the real case K = R and arbitrary
k. Corollary 4.13 (1) Let the determinant of the matrix D2 φ(h)|h=h0 vanish. Assume (k) that R(hj ) → R0 as hj → h0 , hj ∈ Ωh . If the determinant of the matrix ∂12 ψ(h0 + λR0 , λ) . . . ∂1 ∂k ψ(h0 + λR0 , λ) .. .. (4.18) . ∙∙∙ . ∂k ∂1 ψ(h0 + λR0 , λ)
∂k2 ψ(h0 + λR0 , λ)
...
does not vanish for some λ ∈ Rn−k , then there exists a limit A = lim
j→∞
n−k X
λl DRl (hj )
l=1
and (−1) is an eigenvalue of the matrix A. (2) If for h ∈ Ωh there exists λ ∈ Rn−k such that (h, λ) ∈ Ω and if −1 is an eigenvalue Pn−k of the matrix l=1 λl DRl (h), then the determinant of the matrix D2 φ(h) equals zero. (3) If for a global fibration at least one of the matrices DRj (h), 1 ≤ j ≤ n − k, has a non-zero real eigenvalue, then the determinant of the matrix D2 φ(h) equals zero. Proof. (1) First we note that for λ = 0 matrix (4.18) is equal to the singular matrix D2 φ(h0 ) and thus the non-singularity of (4.18) implies λ 6= 0. Differentiating (4.8) with respect to h, we get ∂12 ψ(h + λR(h), λ) . . . ∂1 ∂k ψ(h + λR(h), λ) n−k X .. .. (I + λl DRl (h)) = D2 φ(h). k . ∙∙∙ . ∂k ∂1 ψ(h + λR(h), λ)
...
∂k2 ψ(h + λR(h), λ)
l=1
(4.19)
50
Pn−k The assumption implies then the limit of Ik + l=1 λl DRl (hj ) exists and so the limit Pn−k of l=1 λl DRl (hj ) exists as well in view of λ 6= 0. We also see that Ik + A is singular. This means that −1 is an eigenvalue of A, implying the statement. (3) Suppose that μ ∈ R is an eigenvalue of DRj (h) for some 1 ≤ j ≤ n − k : DRj (h)v = μv. Differentiating (4.8) with respect to h, we get (∂i ∂j ψ(h + λR(h), λ))1≤i,j≤k (Ik +
n−k X
λl DRl (h)) = D2 φ(h).
l=1
Now we apply both sides to v and take λj = −μ−1 and λl = 0 for l 6= j. This leads to D2 φ(h)v = 0. The proof of (2) is similar. Finally, let us make several remarks for the case k = n − 1. In this case there is only one matrix DR1 . If for some h the matrix DR1 (h) does not have any real eigenvalues, then the fibration is extendible globally in λ ∈ R locally uniformly in h. Corollary 4.14 Let k = n − 1 and assume that D2 φ(h)|h=h0 is singular. Let R0 = limj R(hj ) as hj → h0 , hj ∈ Ω(k) h . Then we have . . . ∂1 ∂n−1 ψ(h + λR0 , λ) ∂12 ψ(h + λR0 , λ) .. .. (4.20) det ≡0 . ∙∙∙ . ∂n−1 ∂1 ψ(h + λR0 , λ)
2 ∂n−1 ψ(h + λR0 , λ)
...
for all λ in a neighborhood of zero. Moreover, if there exist the limit A = limj DR(hj ) then (4.20) holds for all |λ| < ρ(A)−1 , where ρ(A) is the spectral radius of A. If the limit limj DR(hj ) does not exist, then (4.20) holds for all λ, for which it is defined.
Proof. If equality (4.20) does not hold then by the second part of Corollary 4.13 there exists the limit A = limj DR(hj ) and −1 is an eigenvalue of the matrix λA. Clearly, this is false for all |λ| < ρ(A)−1 . The last assertion follows from (4.19), if we use that det(In−1 + μA) 6= 0. Note, that λ in Corollary 4.14 can be chosen independent of R0 , since it is continuous as a function of R0 , which takes its values in the compact Grassmannian G1 (Cn ). In the complex notation of Section 3 the statement of Corollary 4.14 is to say that ˉ 0 , 0) ∩ Ω ⊂ Ω\Ω(n−1) , (h0 , 0) ∈ Ω\Ω(n−1) ⇒ κ(h which corresponds to Lemma 3.25.
4.4
Existence of singular fibrations of gradient type
Theorem 3.9 and results of Section 3 show that the smallest dimension for which a singular fibration can exist for K = C, is n = 4 with k = 3. Let us show that in C4 there is a fibration by lines corresponding to a fibration of gradient type for Γ = ∇ψ, and analytic (even polynomial) function ψ. Example 4.15 Let ψ(ξ1 , ξ2 , ξ3 , ξ4 ) = ξ1 ξ22 + (ξ3 − ξ2 ξ4 )2 . 51
The Hessian of the function ψ has the form 0 2ξ2 2 2ξ 2ξ 2 1 + 2ξ4 D2 ψ(ξ) = 0 −2ξ4 0 4ξ2 ξ4 − 2ξ3
0 −2ξ4 2 −2ξ2
with k = 3. Moreover,
0 4ξ2 ξ4 − 2ξ3 −2ξ2 2ξ22
rank D2 ψ|ξ2 =0 = 2, rank D2 ψ|ξ2 =ξ3 =0 = 2, rank D2 ψ|ξ1 =ξ2 =ξ3 =0 = 1. For ξ2 6= 0 the kernel of the matrix D2 ψ(ξ) is one dimensional, ker D2 ψ(ξ) = h(
ξ3 − ξ4 , 0, ξ2 , 1)i. ξ2
Therefore, for λ = ξ4 the fibration, given by the mapping R, has coordinates R1 (h) = h3 2 3 1 h2 , R (h) = 0, R (h) = h2 . The function R has a essential singularity at h2 = h3 = 0 and we have Ωsing = {ξ2 = ξ3 = 0}. Example 4.16 Functions ψ(ξ1 , ξ2 , ξ3 , ξ4 ) = ξ1 ξ2k + (ξ3 − ξ2 ξ4 )m with k, m ≥ 2 lead to fibrations with essential singularities at ξ2 = ξ3 = 0. The fibers are given by m (ξ3 − ξ2 ξ4 )m−1 ker D2 ψ(ξ) = h( , 0, ξ2 , 1)i k ξ2k−1 and R1 (h) =
m hm−1 3 . k hk−1 2
Example 4.17 In n-dimensional space ξ = (x1 , . . . , xn−3 , y, z, w) define ψ(ξ) = y 2
n−3 X i=1
xi + (z − yw)2 .
The level sets of the gradient ∇ψ have dimension n − 3, and hence k = 3. One can check that dim Ωsing = n − 2 (this also follows from dimension estimates of Theorem 3.9 with k = 3). Example 4.18 Consider ψ(x, y, z, v, w, s, t) = xy 2 + sv 2 + (z − yw − vt)2 . The maximal rank of the Hessian D2 ψ equals k = 5 and its kernel is spanned by the vectors z − yw − vt z − yw − vt , 0, 0, y, 1, 0, 0), (0, 0, 0, v, 0, , 1). ( y v For y = 0, z − vt = 0 the first vector has an essential singularity, meanwhile the second vector is continuous at v 6= 0. Hence {y = 0, z = vt} ⊂ Ωsing and dim Ωsing = n − 2 by Theorem 3.23. 52
These examples present singular fibrations for different dimensions of the fibers. On the other hand, the dimension of Ωsing in all examples equals n − 2. In examples 4.15–4.16 it is necessary because for n = 4 Theorem 3.10 implies k = 3. It follows then by Theorem 3.9, (1), that dim Ωsing = n − 2. In this case a stronger Proposition 3.30 holds as well. In Examples 4.17 and 4.18 with different dimension of the fibers we still have dim Ωsing = n − 2. However, our constructions are based on the same idea for the singularity. It would be interesting to investigate whether the condition dim Ωsing = n − 2 is necessary for fibrations of gradient type.
5 5.1
Further estimates for Fourier integral operators Complexification
In this section we will briefly describe the complexification of considered sets. In the complex domain results of Sections 3 and 4 become available. For a real analytic f we denote its extension to a complex domain (complexification). manifold M by M Let Λ be a real analytic conic Lagrangian submanifold of T ∗ M \0. By Theorem 2.3 the generating function φ for a localization Λφ of the manifold Λ is real analytic. The e of the manifold Λ in the complexification of the cotangent bundle complexification Λ ∗ f\0. ] T M is a complex analytic conic Lagrangian submanifold of T ∗ M e e in Let us first assume that variables ξ can be taken as local coordinates for Λ e e e e e 0 , ξ0 ) takes the form (m e 0 , ξ0 ) ∈ Λ. The manifold Λ in a neighborhood of the point (m e 1 ≤ i ≤ dim M f, m e i = Hi (ξ),
where Hi are holomorphic functions, real valued for ξe in the real domain. Lagrangian e is involutive and hence the Poisson bracket defines an ideal on Λ, e i.e. manifold Λ e if f = 0 and g = 0 on Λ, e (cf. [17, 5.4.6]). In particular, {f, g} = 0 on Λ, for all i, j, and, therefore,
e m e e =0 {m e i − Hi (ξ), e j − Hj (ξ)}| Λ ∂Hj ∂Hi f. = , 1 ≤ i, j ≤ dim M e ∂ ξj ∂ ξei
P e ξei vanish. It follows that This means that all mixed derivatives of the form i Hi (ξ)d it is closed and, therefore, exact. We obtain the existence of a holomorphic function e ξ)}. e e then locally assumes the form {(∇H(ξ), H with ∂H/∂ ξei = Hi . The manifold Λ The same argument can be applied to Λ in the real domain to show that Λ is locally parameterized by (∇H(ξ), ξ) with the same function H. The same statement is true in the general setting, when variables ξe do not define a local coordinate system (see [17, 5.6.4] for the complex case and [22, Lemma 3.7.4] for the real case). e be its complexification Let π be the standard projection from T ∗ M to M and let π f to M f. Let Λ(k) be the open subset of Λ, on which the maximal rank k of from T ∗ M the mapping dπ|Λ is attained. By the implicit function theorem the set Σ (k) = π(Λ(k) ) is a smooth analytic k–dimensional submanifold of M contained in Σ = π(Λ). We also g (k) of Σ(k) in M f is a smooth analytic have Λ(k) = N ∗ Σ(k) . The complex extension Σ 53
g (k) is a complex analytic manifold of complex dimension k. Its conormal bundle N ∗ Σ ∗f (k) ∗M . f and T] Lagrangian submanifold of T M containing Λ , where we identify T ∗ M g g g (k) denote a complexification of Λ(k) . The sets Λ (k) and N ∗ Σ (k) are of complex Let Λ ∗g e and the level dimension n and locally coincide. Therefore, the set N Σ(k) is open in Λ sets of π e|Λg are linear subspaces of complex dimension k of cotangent spaces of the (k) g f. The complement Λ\ e Λ(k) is analytic and nowhere dense. manifold M Now let us apply the described construction to the product M = X × Y , where X and Y are two real analytic manifolds. Let Λ be a homogeneous canonical relation satisfying the local graph condition and let φ be its generating function as in (2.10) and (2.11). Then φ is real analytic and by φe we denote its complex extension. Since the complex holomorphic extension is unique, the function φe is a generating function e Reasoning as in Section 2.4 we get that the level sets (fibers) of the mapping for Λ. e y , ξ) e Γ : ξe 7→ ∇ξeφ(e
(5.1)
e in the open dense set where are affine subspaces of Cn of codimension k, for all (e y , ξ) 2 e the rank of Dξeξeφ is maximal, equal to n + k. Thus, we proved the following theorem.
Theorem 5.1 Let X, Y be real analytic manifolds of dimension n and let Λ be an analytic homogeneous canonical relation from T ∗ Y \0 to T ∗ X\0. Let the maximal rank of dπX×Y |Λ be equal to n + k. Let φ be a local generating function for Λ, as in (2.10), e Ye , Λ, e φe be complex analytic extensions of X, Y, Λ, φ, respectively. Then (2.11). Let X, the mapping (5.1) satisfies properties (R1)–(R3) from Introduction. Remark 5.2 Let X and Y be smooth real manifolds, but not analytic. Let φe be an almost analytic extension of a generating function φ. This means that its restriction to the real domain equals φ and ∂ˉφe = 0 in the real domain with infinite multiplicity. In the case Im φe ≤ 0 such theory of Fourier integral operators exists for almost analytic extensions (cf. [38], [33], [4], [36]). In this case results of Section 3 can be applied modulo non-analytic part, for which all the derivatives vanish.
5.2
Operators, commuting with translations
Let X, Y be open sets in Rn . Let an operator T ∈ I μ (X, Y ; Λ) commute with translations. It follows that T is a convolution operator with some distribution w. This distribution w is a Lagrangian distribution of order μ, associated to some conic Lagrangian manifold C ⊂ T ∗ X. The smooth factorization condition means that the rank of the projection π : C → Rn does not exceed k, is equal to k at points in general position, and the projection π can be factored in a composition π = α ◦ β, where β is a fibration from Λ to some k-dimensional smooth manifold M and α is a smooth mapping from M to Rn . The mapping α induces the pullback α∗ from C ∞ (Rn ) to C ∞ (M ) by α∗ f = f ◦ α and the pushforward α∗ from (C ∞ )0 (M ) to (C0∞ )0 (Rn ) by α∗ g = g ◦ α∗ . Let μ be a smooth positive compactly supported measure in M . Then the distribution v = α∗ μ defines a Fourier integral distribution with the Lagrangian 54
manifold C, obtained from Λ by fixing an arbitrary point in T ∗ X. The pullback α∗ maps measures in M to measure in Rn , hence v is also a measure in Rn . In order to see that v = α∗ μ defines a Fourier integral distribution we write v in some local coordinates z in M : RR v(x) = (2π)−n R R eihx−y,ξi v(y)dydξ = (2π)−n R R eihx−y,ξi (α∗ μ)(y)dydξ = (2π)−n R R αy∗ (eihx−y,ξi )(z)μ(z)dzdξ eihx−α(z),ξi μ(z)dzdξ. = (2π)−n As a candidate for the distribution w above we can take the Fourier integral distribution v. One readily checks that the order of v as a Fourier integral distribution is equal to −k/2. For the convolution operator with v we have the estimate ||T f ||Lp = ||v ∗ f ||Lp ≤ v(Rn )||f ||Lp , based on the translation invariance of the Lp norm. Letting p go to one we get the continuity of operators of order −k/2 in Lp spaces, which corresponds to Theorem 2.9.
5.3
Application to Fourier integral operators
Let us introduce the following definition. Definition 5.3 A Fourier integral operator T ∈ Iρμ (X, Y ; Λ) is called analytic if the sets X, Y, Λ are real analytic. Note, that we do not require the analyticity of the symbol in the definition since we are interested in the properties of operators modulo smooth terms. It is also often convenient to assume that the symbol of T is compactly supported. Let a Fourier integral operator T ∈ Iρμ (X, Y ; Λ) with 1/2 ≤ ρ ≤ 1 commute with translations. Here X and Y are open sets in Rn . Invariance means that T is a convolution operator with some distribution. The theory of such operators as Fourier multipliers is well known but their Lp properties require additional study (cf. [42], [60] and the previous Section 5.2). For the case ρ = 1/2 see [27]. The distribution w is a Lagrangian distribution associated with the Lagrangian manifold Λ(w) ⊂ T ∗ (Rn ), obtained by fixing any point in T ∗ Y . By Theorem 2.3 there exists a phase function φ and a generating function H such that Λ(w) is locally equal to Λφ with a positively homogeneous in ξ function φ(z, ξ) = hz, ξi − H(ξ), (5.2) where Λφ = {(∇H(ξ), ξ)}. The Lagrangian distribution w takes the form Z w(z) = eiφ(z,ξ) a(z, ξ)dξ, where a ∈ Sρμ . The operator T assumes now the form Z Z T u(x) = u ∗ w(x) = eiΦ(x,y,ξ) a(x − y, ξ)u(y)dξdy 55
(5.3)
with phase function Φ(x, y, ξ) = hx − y, ξi − H(ξ). We assume that rank dπX×Y |Λ ≤ n + k
(5.4)
for some 0 ≤ k ≤ n − 1 and that the rank n + k is attained somewhere. Condition (3) of Theorem 2.18 means that (5.5) rank D2 H(ξ) ≤ k, for all ξ ∈ Ξ. By Theorem 5.1 the function H has a holomorphic extension, which we also denote by H. Since H is homogeneous of degree one, we can single out a conic variable τ and write ξ as ξ = (θ, τ ) ∈ Cn−1 × C. For ζ ∈ Cn−1 let us define F (ζ) = H(ζ, 1). The homogeneity of H implies H(θ, τ ) = τ F (θ/τ ) and ∇θ H(θ, τ ) ∂τ H(θ, τ ) 2 Dθθ H(θ, τ ) ∂τ ∇θ H(θ, τ ) ∂τ2 H(θ, τ )
= ∇F (θ/τ ), = −h∇F (θ/τ ), θ/τ i + F (θ/τ ), = D2 F (θ/τ )/τ, = −hD2 F (θ/τ ), θ/τ i/τ, = (θ/τ )T D2 F (θ/τ )(θ/τ )/τ.
Writing ζ = θ/τ , we obtain 1 D H(θ, τ ) = τ 2
D2 F (ζ) −ζ T D2 F (ζ)
−D2 F (ζ)ζ ζ T D2 F (ζ)ζ
.
Therefore, rank D2 H(θ, τ ) = rank D2 F (θ/τ ) and conditions (5.4) and (5.5) are equivalent to rank D2 F (ζ) ≤ k.
The mapping F is holomorphic in some open subset Ω of Cn−1 . The kernels of D2 F are the complements to the conic direction in the kernels of D2 H. Theorem 5.1 implies
Theorem 5.4 Let X be an open set in Rn and let Λ be a real analytic conic Lagrangian submanifold of T ∗ X\0. Let H be its generating function, i.e. let Λ be locally of the form (5.2), (5.3). Let k ≤ n − 2 be such that rank D2 H ≤ k. In a neighborhood of ξn = 1 define F by F (ζ) = H(ζ, 1). Then F is real analytic and its holomorphic extension we denote by F : Ω → C as well, where Ω is an open set in Cn−1 . Then the mapping ∇F : Ω ⊂ Cn−1 → Cn−1 satisfies conditions (A1)–(A3) of Introduction. Remark 5.5 Note that condition k ≤ n − 2 in Theorem 5.4 does not restrict the generality because for k = n − 1 the level sets of ∇F are zero dimensional and the smooth factorization condition holds trivially. 56
Having established a connection with Section 3 we can use its results. Theorem 5.6 Let T ∈ Iρμ (X, Y ; Λ) be an analytic Fourier integral operator, commuting with translations. Let μ ≤ −(k +(n−k)(1−ρ))|1/p−1/2|, 1 < p < ∞, 1/2 ≤ ρ ≤ 1. Let Λ be such that rank dπX×Y |Λ ≤ n + k, with0 ≤ k ≤ 2. Then T is continuous from Lpcomp (Y ) to Lploc (X).
The statement follows from Theorems 5.4, 3.10 and Theorem 2.10. Theorem 5.7 Let X, Y be open sets in Rn with n ≤ 4 and let T ∈ Iρμ (X, Y ; Λ) be an analytic Fourier integral operator commuting with translations. Let μ ≤ −(k + (n − k)(1 − ρ))|1/p − 1/2|, 1 < p < ∞ and 1/2 ≤ ρ ≤ 1. Let 0 ≤ k ≤ 3 be such that rank dπX×Y |Λ ≤ n + k. Then T is continuous from Lpcomp (Y ) to Lploc (X). For k = n − 1 and k = 0 the statement follows from Theorem 2.10. For k = 1 it follows from Theorems 5.4 and 5.6. The last case n = 4, k = 2 follows from Theorems 5.4, 3.10 and 2.10. Remark 5.8 The main idea behind the last two theorems is that if X, Y are real analytic and if an analytic conic Lagrangian manifold Λ ⊂ T ∗ X\0×T ∗ Y \0 is associated to an operator commuting with translations, and rank dπX×Y |Λ ≤ 2, then the smooth factorization condition is satisfied. Remark 5.9 Let T satisfy conditions of Theorems 5.6 and 5.7. Then it follows from Theorem 2.11 that T is continuous from H 1 to L1 provided μ ≤ −(k +(n−k)(1−ρ))/2.
6 6.1
Cauchy problem for hyperbolic equations Estimates for the solutions
Results of the previous section can be applied to improve the known Lp estimates for the fixed time solutions of hyperbolic Cauchy problem. For more detailed discussions of the Cauchy problem for hyperbolic partial differential equations we refer to [3], [6], [22], [60], [57]. The class of operators which we consider below contains operators with constant coefficients. Solutions to the Cauchy problem for such operators are well known ([11], [7]). Results of this section have partially appeared in [51] and [54]. Let the operator m X P (t, ∂t , ∂x ) = ∂tm + Pj (t, ∂x )∂tm−j j=1
be strictly hyperbolic and have order m in a compact n-dimensional smooth analytic manifold X with n ≤ 4. Let p(t, τ, ξ) be the principal symbol of the operator P . Strict hyperbolicity means that the polynomial p(t, τ, ξ) has m real distinct roots in
57
τ . The roots τj (t, ξ) are real, homogeneous of degree one in ξ and smooth in t. The corresponding Cauchy problem is the equation P u(t, x) = 0, t 6= 0, (6.1) ∂tj u|t=0 = fj (x), 0 ≤ j ≤ m − 1. The principal symbol p (or σP ) of the operator P can be written as a product p(t, τ, ξ) =
m Y
j=1
(τ − τj (t, ξ)).
For small t a solution to the Cauchy problem (6.1) can be obtained in the following way. Let Φj (t, x, ξ) solve the eikonal equation: ∂ ∂t Φj (t, x, ξ) = τj (t, ∇x Φj ), (6.2) Φj (t, x, ξ)|t=0 = hx, ξi. Under the strict hyperbolicity condition the Cauchy problem (6.1) is well posed and modulo a smooth term its solution u(t, x) can be obtained as a finite sum of elliptic Fourier integral operators, smoothly dependent on t: u(t, x) =
m m−1 X XZ j=1 l=0
Rn
Z
Rn
e2πi(Φj (t,x,ξ)−hy,ξi) ajl (t, x, ξ)fl (y)dξdy.
(6.3)
Canonical relations are locally generated by function Φj from (6.2), and symbols ajl ∈ S −l can be found from additional transport equations [5], [22]. Expression (6.3) is smooth in t. Let f be a smooth function on an open set U in T ∗ X. Then df is a smooth one-form. A smooth vector field Hf is called the Hamiltonian vector field for f if the equality σ(v, Hf ) = hv, df i = v(f ) holds for all smooth vector fields v. In local coordinates at (x, ξ) ∈ U the vector field Hf takes the form Hf =
n X ∂f ∂ ∂f ∂ − . ∂ξ ∂x ∂x j j j ∂ξj j=1
For a fixed value of t the canonical relation of the solution operator in (6.3) has the form m [ Ct = {(x, ξ, y, η) : χt,j (y, η) = (x, ξ)}, j=1
where canonical transformations χt,j : T ∗ X\0 → T ∗ X\0 are the flows from (y, η) at t = 0, passing through (x, ξ) at t, along the Hamiltonian vector field X n n X ∂τj ∂ ∂τj ∂ ∂τj ∂ Hj = − = ∂x ∂ξ ∂ξ ∂x ∂ξj ∂xj j j j j j=1 j=1 in T ∗ X\0. Regarding t as one of the variables, we can write the canonical relation of the operator (6.3) as C=
m [
j=1
{(x, ξ, y, η, t, τ ) : τ = τj (t, ξ), χt,j (y, η) = (x, ξ)}. 58
2 Using the representation and the asymptotics of Dξξ Φj with respect to t, one can show 2 ([56]) that the maximal rank Dξξ Φj = 2n − 1 under an additional assumption that for all t, except possibly a discrete set in R, at least one of the roots τj is elliptic. In particular, this assumption holds for the wave equation with variable coefficients. In 2 Φj , it is possible to apply results general, if k is the maximal rank of the matrices Dξξ p of Section 5.3 to obtain sharp L estimates for the solutions of the Cauchy problem (6.1). Let us first give some examples of hyperbolic operators for which the rank of the Hessian of a phase function of the solution operator equals k < n−1. For the simplicity we restrict our examples to the case of constant coefficients. As usual, let k denote the maximal rank of the Hessians. We consider now Cauchy problem (6.1) in R1+n with n ≥ 3. Let a second order operator P be defined by
p(∂t , ∂ξ ) = ∂t2 −
l+1 X
∂ξ2j ,
j=1
plus lower order terms, and 0 ≤ l ≤ n − 1. In this case the roots τj are τ1 = −τ2 = Pl ( j=1 ξj2 )1/2 and the maximal rank k = l. The order m of an operator P can be an arbitrary m ≥ 2 and the rank k can take an arbitrary value between zero and n − 1. Indeed, let the principal symbol of an operator P be equal to m/2
Y l=1
(τ 2 −
k X j=1
2 ξj2 − lξk+1 )
for even m, and for odd m we multiply it by τ − ξ1 , then the rank for such operator will be equal to k. If we replace ξ1 and ξ2 by other ξi , we also get operators involving derivatives with respect to all variables. Theorem 6.1 Let u(t, x) solve the Cauchy problem (6.1) and let Φj (t, x, ξ) solve the eikonal equation (6.2). Let k = max rank x,ξ,j
∂2 Φj (t, x, ξ) ∂ξ 2
and let αp = k|1/p − 1/2|, 1 < p < ∞. Let the Cauchy data satisfy fj ∈ Lpα+αp −j (X). Then u(t, ∙) ∈ Lpα (X). Moreover, if k=
max
x,ξ,j,t∈[−T,T ]
rank
∂2 Φj (t, x, ξ) ∂ξ 2
for a fixed 0 < T < ∞, then ||u(t, ∙)||Lpα ≤ CT
m−1 X j=0
||fj ||Lpα+α
The orders α + αp − j can not be improved.
59
p −j
, t ∈ [−T, T ].
(6.4)
Proof. Let us look for the solutions to the eikonal equation (6.2) in the form Φj (t, x, ξ) = hx, ξi − Hj (t, ξ). Then equations (6.2) reduce to a Cauchy problem for ordinary differential equations ∂Hj (t, ξ) = −τj (t, ξ), ∂t
(6.5)
with Hj (0, ξ) = 0, where ξ is a parameter. Because p is strictly hyperbolic and analytic in ξ, τj and Φj must be analytic in ξ and smooth in t. The roots τj are homogeneous of degree one in ξ and hence functions Hj (t, ξ) are analytic and homogeneous of degree one in ξ for small t. Estimates (6.4) now follow from Theorem 5.7. The sharpness follows from Theorem 7.1. Remark 6.2 The statement of Theorem 6.1 can be generalized to other dimensions n of X. Under an additional assumption that k ≤ 2, solution u(t, ∙) belongs to Lpα and sharp estimates (6.4) are valid. The proof is similar to the proof of Theorem 6.1 and is based on Theorem 5.6. If X is not compact, the statement of Theorem 6.1 holds in Lpcomp . In other function spaces we have the following statement. Theorem 6.3 In conditions of Theorem 6.1 the following estimates hold. (1) Let 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞. Then ||u(t, ∙)||
Lqα
≤ CT
m−1 X j=0
||fj ||Lpα+α
pq −j
, t ∈ [−T, T ],
(6.6)
where αpq = −n/p + k/q + (n − k)/2. (2) In Lipschitz spaces Lip(γ) holds ||u(t, ∙)|| Lip (α) ≤ CT
m−1 X j=0
||fj || Lip (α+k(n−1)/2−j) , t ∈ [−T, T ].
(6.7)
The orders in the above statements can not be improved. The estimates follow from corresponding theorems for Fourier integral operators from the previous section. The sharpness of the estimates will be considered in the following section. The best Lp regularity properties are exhibited by operators, for which the order αp in Theorem 6.1 equals zero. In this case the regularity properties of solutions are the same as for elliptic Cauchy problems, for which the solutions are given by pseudodifferential operators. This has an underlying explanation. The condition αp = 0 implies k = 0 and in the next section we will show that solution operators can be obtained from pseudo-differential operators by a composition with Fourier integral operators induced by smooth coordinate changes. Anticipating the next section we give now a statement for this situation. 60
Theorem 6.4 Let n ∈ N, 1 < p < ∞ and p 6= 2. For the Cauchy problem (6.1) the following conditions are equivalent: (1) The principal symbol of the operator P has the form σP (t, τ, ξ) =
m Y
j=1
(τ − τj (t, ξ))
(6.8)
where τj (t, ξ) are linear in ξ. (2) For any Cauchy data fj ∈ Lpm−j the solution u(t, ∙) of the Cauchy problem (6.1) belongs to Lpm . In both cases there exist pseudo-differential operators Qjl , Sjl ∈ Ψ−l (Y ) and diffeomorphisms σj : X → Y such that the solution to the Cauchy problem (6.1) has the form m m m−1 m−1 XX XX (σj∗ ◦ Qjl )fl = (Sjl ◦ σj∗ )fl , (6.9) u= l=0 j=1
σj∗
where αp = 0.
l=0 j=1
are the pullbacks induced by σj . Sobolev space estimates (6.4) hold with
For p = 2 condition (2) holds for arbitrary operators P . This is why it is necessary to exclude the case p = 2 from the formulation of the theorem. The proof will be given in Section 7.
6.2
Monge–Amp` ere equation
Generating functions φ satisfy the following parametric Monge–Amp`ere equation: det
∂2φ (y, ξ) = 0 ∂ξ 2
(6.10)
for all (y, ξ) ∈ Y × Ξ. Let us consider an invariant version of equation (6.10): det
∂2φ (ξ) = 0, ∀ξ ∈ Ω, ∂ξ 2
(6.11)
which corresponds to operators commuting with translations. Such equations are called the simplest Monge–Amp`ere equations, cf. [34, 8.2]. Note that if φ is convex, equation (6.11) has the unique solution, which is zero. In general, the problem det
∂2φ (ξ) = f ∂ξ 2
(6.12)
almost everywhere in the ball B = {ξ ∈ Rn : |ξ| < 1}, n ≥ 2, with zero boundary conditions on ∂B and non-negative function f ∈ C 2 (B), has the unique solution in ˉ for all α ∈ (0, 1), B in the class of convex functions. Solution φ belongs to C 3+α (B) ˉ (cf. [34, 8.2.2]). In our case we have f = 0. Let φ solve equation provided f > 0 on B. (6.11). We do not require the convexity now. Using Theorems 3.9 and 3.10 we conclude Corollary 6.5 Let Ω be an open set in Rn (or Cn ) and let φ(ξ) be an analytic solution to the Monge–Amp`ere equation (6.11) in Ω, such that the level sets of ∇φ(ξ) are affine 2 for all ξ with the maximal rank ∂∂ξφ2 (ξ) ≤ 2. Then the fibration by the level sets of ∇φ(ξ) allows an analytic extension to Ω. 61
7
Sharpness of the estimates
In this section we will discuss the sharpness of the orders of continuous Fourier integral operators for different ranks of projections. Let X and Y be smooth paracompact n–dimensional manifolds and let T ∈ Iρμ (X, Y ; Λ) be a Fourier integral operator with a canonical relation Λ. In the sequel we will concentrate on the essentially homogeneous case (ρ = 1). However, for 1/2 ≤ ρ < 1, the sharpness of the order μ = −(n − ρ)|1/p − 1/2| for the continuity in Lp can be shown by the following example. Let Σ be a manifold given by the set of points (x0 , x00 , y 0 , y 00 ) with x0 = y 0 and |x00 − y 00 | = 1. Let Λ0 be the conormal bundle N ∗ Σ. Let Ψ(ξ) be a smooth function, homogeneous of degree zero for large ξ, supported in the truncated cone {ξ = (ξ 0 , ξ 00 ) ∈ Rk+1 × Rn−k−1 : |ξ 0 | ≥ 1, |ξ 00 |/2 ≤ |ξ 0 | ≤ 2|ξ 00 |}, and equal to one in an open truncated subcone. Let T be a convolution operator with the kernel Z 1−ρ 0 K(x, y) = Ψ(ξ)ei|ξ| |ξ|μ ei[hx−y,ξi+|ξ |] dξ.
One can readily check that T ∈ Iρμ (Rn , Rn ; Λ). Let gˆσ (ξ) = (1 + |ξ|2 )−σ/2 . Then gσ ∈ Lp for σ > n(1 − 1/p). Acting by T on gσ and applying the stationary phase method in polar coordinates for ξ 0 , one can show that T gσ 6∈ Lploc for σ converging to n(1 − 1/p), and μ > −(n − ρ)|1/p − 1/2|. Asymptotic expansions for such operators were derived in [67] and the analysis was used in [56] to show that the order μ is sharp in Theorem 2.10.
7.1
Essentially homogeneous case
In the essentially homogeneous case with ρ = 1 one can show that the order μ = −k|1/p − 1/2| is sharp for arbitrary Lp continuous elliptic operators satisfying the condition (7.1) rank dπX×Y |Λ ≤ n + k.
It turns out that in order to check that an operator is not continuous in Lp it suffices to let T act on functions f with point singularities in Lp . In this case the singularities of T f can appear only in directions transversal to some fixed k–dimensional manifold in X.
Theorem 7.1 Let the canonical relation Λ be a local canonical graph such that the inequality rank dπX×Y |Λ ≤ n + k holds with 0 ≤ k ≤ n − 1, and the rank n + k is attained somewhere. Then elliptic operators T ∈ I μ (X, Y ; Λ) are not bounded from Lpcomp (Y ) to Lploc (X), provided μ > −k|1/p − 1/2| and 1 < p < ∞. Let us note at once that by the equivalence-of-phase-function theorem it is sufficient to consider operators T in Rn with kernels, locally given by Z K(x, y) = ei[hx,ξi−φ(y,ξ)] b(x, y, ξ)dξ, (7.2) Rn
with symbols b ∈ S μ supported in compact sets with respect to x and y. The local graph condition means that the phase function φ satisfies det φ00yξ 6= 0 62
(7.3)
on the support of b. Locally Λ has the form {(∇ξ φ, ξ, y, ∇y φ)}. We can assume that 1 < p ≤ 2. For 2 < p it suffices to consider the adjoint operator T ∗ and the statement follows from the result for the conjugate index p0 = p/(p − 1) < 2. The set Λ0 = {λ ∈ Λ : rank dπX×Y |Λ (λ) = n+k} is not empty and is open in Λ. Let λ0 = (x0 , ξ0 , y0 , η0 ) ∈ Λ0 . Consider a family of functions fs (y) = ((I − Δ)−s/2 δy0 )(y) for a fixed value of y0 ∈ Y . Let K−s be the integral kernel of (I − Δ)−s/2 . Then we have Z fs (y) = K−s (y, z)δy0 (z)dz = K−s (y, y0 ).
Using standard estimates for Schwartz kernels of pseudo-differential operators ([13], [22], [60]), we get that |K−s (y, y0 )| ≤ C|y − y0 |−n+s in some local coordinate system. This means that fs ∈ Lploc if and only if s > n(1 − 1/p). Let Σ = πX×Y (Λ ∩ U ), where U ⊂ Λ0 is a neighborhood of λ0 . Because the rank of πX×Y is constant in U , the set Σ ⊂ X × Y is a smooth k–dimensional manifold, given by equations hj (x, y) = 0, 1 ≤ j ≤ n − k, in a neighborhood of y0 . The vectors ∇h1 , . . . , ∇hn−k are linearly independent on Σ. Then, microlocally, Λ is the conormal bundle of Σ, and the phase function of the operator T assumes the form Φ(x, y, λ) =
n−k X
λj hj (x, y).
j=1
Since a composition with a pseudo-differential operator does not change the canonical relation, we get that T ◦ (I − Δ)−s/2 ∈ I μ−s (X, Y ; Λ). It follows that T fs (x) = T ◦ (I − Δ)−s/2 (δy0 )(x), which in local coordinates can be written as P R R ˉ y (y)dλ)dy ˉ T fs (x) = Rn ( Rn−k ei λj hj (x,y) a(x, λ)δ 0 R ˉ h(x,y ˉ ihλ, ˉ ˉ 0 )i (7.4) a(x, λ)dλ = Rn−k e ˉ y0 )), = (2π)n−k (F −1 a)(x, h(x, ˉ and h ˉ are vectors with components λj and hj , respectively, and F −1 is the where λ inverse Fourier transform. The symbol a ∈ S μ−s+k/2 (Rn−k ) is obtained from the symbol of the operator T ◦ (I − Δ)−s/2 by using the stationary phase method, where we can eliminate k variables. The inverse Fourier transform of the symbol a with respect to the second variable is Z (2π)n−k (F −1 a)(x, ζ) = eihλ,ζi a(x, λ)δˆ0 (λ)dλ = P δ0 (ζ) = K(ζ, 0), Rn−k
where P ∈ Ψm (Rn−k ) is a pseudo-differential operator of order m = n − s + k/2 with symbol a(x, λ) ∈ S m and K is the integral kernel of P . The function K(ζ, 0) is equivalent to |ζ|−(n−k)−m . For the set Σy0 = {x : (x, y0 ) ∈ Σ} we have dist(x, Σy0 ) ≈ ˉ y0 )|, and, therefore, |h(x, ˉ y0 )) ∼ |dist(x, Σy )|−(n−k)−(μ−s+k/2) , (2π)n−k (F −1 a)(x, h(x, 0 locally uniformly in x. It follows from (7.4) that the function T fs is smooth in Σy0 . Hence T fs 6∈ Lploc (Rn ) if and only if p(n − k + μ − s + k/2) ≥ n − k. 63
(7.5)
For s it means that s ≤ μ + (n − k)(1 − 1/p) + k/2. Thus, in order to have fs ∈ Lploc and T fs 6∈ Lploc , it is sufficient to have the inequality n(1 − 1/p) < μ + (n − k)(1 − 1/p) + k/2, which is equivalent to μ > −k|1/p − 1/2|. Concerning Lp − Lq continuity we have: Corollary 7.2 Let T and Λ satisfy conditions of Theorem 7.1 and let 1 < p, q < ∞. Then T is not bounded from Lpcomp (Y ) to Lqloc (X), provided μ > n(1/q − 1/p) − k(1/q − 1/2). To prove this we can apply the same argument as in the proof of Theorem 7.1. The only difference is that now we do not need to assume p ≤ 2 and inequality (7.5) is replaced by q(n − k + μ − s + k/2) ≥ n − k. Remark 7.3 In the case when k = n−1 and 1 < p ≤ q ≤ 2, the statement of Corollary 7.2 complements Theorem 2.12, (2). By duality we also have a statement, analogous to the one in Theorem 7.6, for indices 2 < p ≤ q < ∞. Remark 7.4 The operator T in Theorem 7.1 is not bounded as a linear operator in Sobolev spaces Lpα → Lpα−k|1/p−1/2|−μ , 1 < p < ∞.
7.2
A representation formula for continuous operators of small negative orders
It is well known that pseudo-differential operators of zero order are continuous in Lp for 1 < p < ∞ (cf. Section 2.3). Now we will show that all Lp continuous elliptic Fourier integral operators can be obtained from pseudo-differential operators by a composition with operators, induced by a smooth coordinate change. A smooth mapping σ : X → Y induces the pullback σ ∗ : C ∞ (Y ) → C ∞ (X), defined by (σ ∗ f )(x) = f (σ(x)). It is not difficult to see that σ ∗ is a Fourier integral operator with phase function hσ(x) − y, ηi. The canonical relation of σ ∗ is equal to the graph of the mapping σ e : T ∗ X\0 → T ∗ Y \0, t −1 where σ e(x, ξ) = (σ(x), −( Dσx ) (ξ)).
Theorem 7.5 Let 1 < p < ∞, p 6= 2, and 0 ≥ μ > −|1/p − 1/2|. Let Λ be a local canonical graph. Then an elliptic operator T ∈ I μ (X, Y ; Λ) is continuous from Lpcomp (Y ) to Lploc (X) if and only if there exist pseudo-differential operators P ∈ Ψμ (X), Q ∈ Ψμ (Y ) such that T = P ◦σ ∗ = σ ∗ ◦Q, where σ ∗ is the pullback by a smooth coordinate change from X to Y . Proof. The pullback σ ∗ is continuous in Lp . Pseudo differential operators P and Q of order μ ≤ 0 are continuous in Lp . Therefore, T is also continuous in Lp . Conversely, let k be such that n+k = maxλ∈Λ rank dπX×Y |Λ (λ). Then n−k is the maximal dimension of the set Σ = πX×Y (Λ) ⊂ X × Y . In view of Theorem 7.1 and Lp continuity of T , it is necessary that k = 0. This means that rank dπX×Y |Λ ≡ n and Σ is a smooth n– dimensional submanifold of X ×Y . The rank of the differential dπX |Σ of the projection 64
πX : X × Y → X is equal to n because Λ is a local canonical graph. Surjectivity of dπX |Σ and condition dim Σ = n imply that πX |Σ is a diffeomorphism and Σ can be locally parameterized by Σ = {(x, σ(x))}, for some diffeomorphism σ : X → Y . It follows that the canonical relation of the pullback σ ∗ is the conormal bundle of Σ, ∗ which is Λ. Therefore, an operator Q in T = σ+ ◦ Q must be pseudo-differential, since its canonical relation is the conormal bundle to the diagonal in X × Y . A similar argument in Y implies that the formula in the theorem holds for an operator P with the same mapping σ, since the canonical relation of σ ∗ equals Λ. Theorem 7.6 Let 1 < p ≤ q < 2 and −n(1/p − 1/q) ≥ μ > −(1/q − 1/2) − n(1/p − 1/q). Let T ∈ I μ (X, Y ; Λ) be elliptic and let Λ be a local canonical graph. Then T is continuous from Lpcomp (Y ) to Lqloc (X) if and only if there exist pseudo-differential operators P ∈ Ψμ (X), Q ∈ Ψμ (Y ), such that T = P ◦ σ ∗ = σ ∗ ◦ Q, where σ ∗ is the pullback by some diffeomorphism σ : X → Y . The continuity of T follows from the continuity from Lp to Lq of pseudo-differential operators of order −n(1/p − 1/q) (Proposition 2.16). Note also that Corollary 7.2 with k = n implies the sharpness of the orders in Proposition 2.16 for elliptic operators P . Finally, let us prove Theorem 6.4. Proof of Theorem 6.4. Assume that the homogeneous part of the highest degree of operator P is as in (6.8). We can differentiate equation (6.5) twice with respect to ξ to 2 2 2 ∂t Hj (t, ξ) = 0. Therefore, ∂ξξ Hj (t, ξ) = ∂ξξ Hj (0, ξ) = 0, and Hj are linear obtain ∂ξξ in ξ. The rank k in Theorem 6.1 is then equal to zero, which implies the second part of the theorem, since pseudo-differential operators of order −j are continuous from (Lpα )comp to (Lpα+j )loc . Conversely, let P be as in the second part of the theorem. If Tl in (6.3) is an operator of order −l, we can apply Theorem 7.5 to the operator T = Tl ◦ (I − Δ)l/2 to obtain 2 2 formula (6.9). Moreover, we have k = 0, and hence ∂ξξ τj (t, ξ) = −∂t ∂ξξ Hj (t, ξ) = 0. Therefore, τj must be polynomial of degree ≤ 1 in ξ. In fact, they are linear, since they are also homogeneous of order one. The Sobolev space estimates follow from the continuity of pseudo-differential operators of order −j from (Lpα )comp to (Lpα+j )loc .
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