THE LIE GROUP OF FOURIER INTEGRAL OPERATORS * AND APPLICATIONS TO FLUID DYNAMICS
Rudolf Schmid Department of Mathematics Emory University Atlanta, GA 30322, USA
[email protected] www.mathcs.emory.edu/∼rudolf
*The Lie Group of Fourier Integral Operators on open Manifolds, to appear (with J.Eichhorn): Comm. in Analysis and Geometry, 2001
1
The theory of pseudodifferential and Fourier integral operators on compact manifolds is well established and their applications in mathematical physics well known [?], [?], [?] . For open (non compact) manifolds this is not the case, and that’s what I would like to focus on in this paper. We are interested in the geometry of of the spaces of pseudodifferential and Fourier integral operators and their associated diffeomorphism groups. Unfortunately, the theory on open manifolds is very technical and much more complicated than the compact case . I will give here only a overview of the results and refer for technicalities to the published papers [4] and [5]. Instead of going through the details of the construction of the Lie group structures of pseudodifferential and Fourier integral operators on open manifolds I will first review the compact case and then explain what goes wrong in the non compact case and how we fixed these problems. Review COMPACT case: In 1985 we proved the following theorem [1],[2],[3]: Theorem: (M.Adams, T.Ratiu, R.Schmid) : The group F IO∗ (M ) of invertible Fourier integral operators on a compact manifold M is a graded ∞-dim Lie group with graded ∞-dim Lie algebra ΨDO(M ) of pseudodifferential operators on M . F IO∗ (M ) is and ∞-dim principal fiber bundle over the base manifold Dif fθ (T˙ ∗ M ) of contact transformations of T˙ ∗ M with gauge group ΨDO∗ (M ) of invertible pseudodifferential operators. ΨDO∗ (M ) −→
F IO∗ (M ) |
↓ Dif fθ (T˙ ∗ M ) In [5] we proved an analogue theorem for open manifolds, but let me first explain the construction of this principal fiber bundle and Lie group in the compact case. What are Fourier integral operators ? Let me call them F IO for short. • F IO generalize pseudodifferential operators (let me denote them by ΨDO) which in tern are a generalization of differential operators (call them DO). So we have as sets the following inclusions F IO ⊃ ΨDO ⊃ DO 2
Let me start to explain ΨDO. P is a classical ΨDO of order m (P ∈ ΨDOm ) if it is locally of the following form : for u ∈ Cc∞ (M ) −n
P u(x) = (2π)
Z Z
ei(x−y)·ξ p(x, ξ)u(y)dy dξ
(1)
where p(x, ξ) classical symbol of order m, i.e. it is a smooth function having an asymptotic expansion p(x, ξ) ∼
−∞ X
pj (x, ξ) ,
(2)
j=m
where each term pj (x, ξ) is homogeneous of degree j in ξ, i.e pj (x, tξ) = tj pj (x, ξ) , t > 0. The leading term am (x, ξ) is called the principal symbol of the operator P and it is globally defined on T ∗ M . For more general symbol classes see eg. [?]. We restrict ourselves to classical ΨDOs. • Special case: If the symbol is a polynomial in ξ i.e. of the form p(x, ξ) =
X
pα (x)ξ α
(3)
|α|≤m
then P is a differential operator of order m, so ΨDO ⊃ DO . ΨDO are oscillatory integrals and highly singular but they are nice operators in the sense that: (i) they are invariant under diffeomorphisms ⇒ they are defined on the whole manifold M as bounded linear operators P : C ∞ (M ) → C ∞ (M ) (ii) they are pseudolocal i.e they preserve the singular support sing supp (P u) ⊂ sing supp (u), and they preserve the wave front sets W F , W F (P u) ⊂ W F (u) (iii) they extend as bounded linear operators to Sobolev spaces P : Hcs (M ) → Hcs−m (M ) (iv) they are closed under composition and the order is additive , i.e. if P ∈ ΨDOm and Q ∈ ΨDOn then P ◦ Q ∈ ΨDOm+n S (v) the space of all pseudodifferential operators of all orders ΨDO = m ΨDOm is an ∞-dim graded Lie algebra, i.e. if P ∈ ΨDOm and Q ∈ ΨDOn then the commutator [P, Q] = P ◦ Q − Q ◦ P ∈ ΨDOm+n−1 . Remark: The principal symbol of the commutator [P, Q] is given by {pm , qn } the canonical Poisson bracket on T ∗ M of the principal symbols 3
pm of P and qn of Q. This property leads to a quantization theory via ΨDO ! Note: The space ΨDO1 of all ΨDO of order one is itself an ∞-dim Lie algebra. QUESTION: Are there LIE GROUPS corresponding to these ∞-dim Lie algebras ? i.e. are there ∞-dim Lie groups G , Go which have ΨDO resp. ΨDO1 as their Lie algebras? Warning: The classical Lie theorems do not hold in ∞ dimensions ! i.e. not every ∞-dim Lie algebra automatically has a corresponding Lie group ! Answer: YES ! G = F IO∗ and Go = (F IOo )∗ the groups of invertible Fourier integral operators (order zero). Let me now explain what F IO are: A ia a (classical) Fourier integral operator of order m (A ∈ F IOm ) if it is locally of the following form : for u ∈ Cc∞ (M ) −n
Au(x) = (2π)
Z Z
eiϕ(x,y,ξ) a(x, ξ)u(y)dy dξ
(4)
where a(x, ξ) is classical symbol of order m as in (2) i.e. a(x, ξ) ∼ j=m aj (x, ξ) and ϕ(x, y, ξ) is a nondegenerate phase function. One of the basic properties of the phase function ϕ(x, y, ξ) is that it is homogeneous of degree one in ξ and that it generates a conic Lagrangian submanifold Λ in T ∗ (M × M ) − 0. We restrict ourselves to those F IO for which this Lagrangian Λ submanifold is the graph of a diffeomorphism η : T˙ ∗ M → T˙ ∗ M , (T˙ ∗ M = T ∗ M − 0), i.e. Λ = graph(η) . This means that the phase function ϕ is locally generating a homogeneous canonical transformation η : T˙ ∗ M → T˙ ∗ M : η ∗ ω = ω, η(tα) = tη(α) which is equivalent that η preserves θ, the canonical 1-form on T ∗ M , i.e. η ∗ θ = θ. These diffeomorphisms are called quantomorphisms. • Special case: if ϕ(x, y, ξ) = (x − y) · ξ then Λ is the diagonal in T˙ ∗ M × T˙ ∗ M and A ∈ ΨDOm , i.e. F IO ⊃ ΨDO ⊃ DO. P−∞
Again like ΨDO the F IO are nice operators (i) they are invariant under diffeomorphisms ⇒ they are defined on the whole manifold M as bounded linear operators A : C ∞ (M ) → C ∞ (M ) 4
(ii) A : C ∞ (M ) → C ∞ (M ) extends as bounded linear operator to Sobolev spaces A : Hcs (M ) → Hcs−m (M ) (iii) they move the wave front sets (the complement of where u is C ∞ ) by a canonical relation Λ, i.e. W F (Au) ⊂ Λ ◦ W F (u), where Λ is the conic Lagrangian submanifold Λ ⊂ T˙ ∗ M × T˙ ∗ M locally generated by the phase function ϕ(x, y, ξ) of A. (iv) they are closed under composition and the order order is additive i.e. if A1 ∈ F IOm and A2 ∈ F IOn then A1 ◦ A2 ∈ F IOm+n . Moreover if η1 is generated by the phase function of A1 and η2 is generated by the phase function of A2 then η1 ◦ η2 is generated by the phase function of A1 ◦ A2 . If A ∈ F IOm is invertible with canonical transformation η then η −1 is generated by the phase function of A−1 ∈ F IO−m . Example: Let f : M → M be a diffeomorphism. Then ∗
−n
f u(x) = (2π)
Z Z
ei(f (x)−y)·ξ u(y)dy dξ
(5)
defines a F IO f ∗ : C ∞ (M ) → C ∞ (M ) whose phase function generates the canonical cotangent lift T ∗ f : T˙ ∗ M → T˙ ∗ M . Let’s denote F IO the space of all Fourier integral operators of all orders S , F IO = m F IOm and F IO∗ the group of all invertible F IO, the group operation being composition and ΨDO∗ the group of invertible ΨDO and let Dif fθ∞ = {η ∈ Dif f ∞ (T˙ ∗ M ) | η ∗ θ = θ} denote the diffeomorphism group of quantomorphisms. We have an exact sequence of groups j
p
I → ΨDO∗ ,→ F IO∗ → Dif fθ∞ → id
(6)
The first map j is just the inclusion, the second map p is defined as follows: For A ∈ F IO∗ , p(A) = η ∈ Dif fθ∞ such that η is generated by the phase function of A . This map p is surjective and its kernel ker p = ΨDO∗ , since the phase function ϕ(x, y, ξ) = (x − y) · ξ generates the identity map id : T˙ ∗ M → T˙ ∗ M . Hence the sequence (6) is and exact sequence of groups. We want to make this into an exact sequence of LIE GROUPS ! Since the order is additive under composition (group operation) the zero order operators also form groups. Let’s consider the corresponding exact 5
sequence of zero order groups and their ”Lie algebras” j
p
Groups
I → (ΨDO0 )∗ ,→ (F IO0 )∗ → Dif fθ∞ → id
(7)
Lie algebras
0 −→ ΨDO0 ,→ ΨDO1 −→ V ec∞ θ → 0
(8)
Let me give some heuristic arguments why these are the corresponding ”Lie algebras”. The formal Lie algebras are identified with the corresponding tangent spaces at the identity. Since the group (ΨDO0 )∗ is an open set in the vector space ΨDO0 , its tangent space at I is isomorphic abelian Lie group ΨDO0 . For the ”Lie algebra” of (F IO0 )∗ , consider a curve in (F IO0 )∗ through the identity I. Taking its derivative (at I) we obtain a ΨDO of order one, since the phase functions of the F IO are homogeneous of degree one, i.e by the chain rule we’r adding to the symbol of an F IO of order zero a term of order one and evaluating at I gives a ΨDO of order one. For Dif fθ∞ , the infinitesimal condition of η ∗ θ = θ is LX θ = 0, i.e. the Lie algebra of Dif fθ∞ is V ec∞ θ = {X vector f ield | LX θ = 0}. Remark: Note that we also have the corresponding exponential maps form the Lie algebras into the groups, but in infinite dimensions they don’t locally generate the Lie groups (unlike in finite dim) i.e we cannot obtain our desired Lie group structures by exponentiating the corresponding Lie algebras (ΨDO0 )∗ 6= exp(ΨDO0 ) , (F IO0 )∗ 6= exp(ΨDO1 ) , Dif fθ∞ 6= exp(V ec∞ θ ) We have to construct these Lie groups and the idea is the following: Note that a ΨDO is determined by its symbol and these form a linear infinite dim. vector space, whereas a F IO is characterized by a symbol and a phase function, the symbols again form a linear space but the phase functions are associated to diffeomorphisms, which form an infinite dim (non linear ) manifold. Idea: Construct an ∞-dim principal fiber bundle such that • base space = Dif fθ∞ (T˙ ∗ M ) • total space = (F IO0 )∗ • fiber = p−1 (η) ' (ΨDO0 )∗ = gauge group This construction is done in 7 steps (see [1],[2],[6])
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• Step 1: We show that Dif fθ∞ = lim Dif fθs is an ILH Lie group, ∞←s (inverse limit of Hilbert Lie groups). On each group of Sobolev class H s diffeomorphisms Dif fθs we construct a Hilbert Lie group structure and take the inverse limit as s → ∞ to obtain Dif fθ∞ = lim Dif fθs , [6]. ∞←s There is one complication here. In order to show that Dif f s θ is a submanifold of Dif f s (T˙ ∗ M ) we need to show that Dif f s (T˙ ∗ M ) is a manifold in the first place, but T˙ ∗ M is never compact, even if M is. So we go to the cosphere bundle S(T ∗ M ) ⊂ T ∗ M , which is compact and carries an induced contact form θS . Each η : T˙ ∗ M → T˙ ∗ M with η ∗ θ = θ is in one to one correspondence with a pair (ηS , h) , where ηS : S(T ∗ M ) → S(T ∗ M ), ηS∗ θS = hθS and h : S(T ∗ M ) → R a smooth function. We showed in [6] that Dif fθs is isomorphic to the semidirect product Dif fθs (S(T ∗ M )) := {(ηS , h) ∈ Dif f s (S(T ∗ M )) >Cs (S(T∗ M)) | ηS∗ θS = hθS } . With this we show that Dif fθ∞ = lim Dif fθs is an ILH Lie group. ∞←s
• Step 2: We show that (ΨDO0 )∗ = ∞←s lim (ΨDO0s )∗ is an ILH Lie group . The topology on ΨDO is determined by the symbols, which are smooth functions and form a vector space which we complete in corresponding Sobolev H s topologies to obtain for the invertible elements (ΨDO0s )∗ Hilbert Lie group structures. Again we take the inverse limit to obtain (ΨDO0 )∗ = lim (ΨDO0s )∗ , [6]. ∞←s
• Step 3: Now we have Lie group structures on both ends of our exact sequence (7) and we can piece these together via a local section σ of the exact sequence (7) σ : U ⊂ Dif fθ∞ → (F IO0 )∗ (9) This requires to construct a global writing of F IOs which are associated to diffeomorphisms η ∈ Dif fθ∞ which are close to the identity. This is quite elaborate and done in [2]. There we define the local section σ as follows: Let η ∈ Dif fθ∞ close to the identity and define σ(η) by −n
σ(η)u(x) := (2π)
Z
Z
Tx∗ M
Bδ (x)
χ(x, y)eiϕH (αx ,y) u(y)| det exp∗ |dydξ
(10)
where ϕH (αx , y) = ϕ0 (αx , y) + H(αx ), with ϕ0 (αx , y) = αx · exp−1 x (y) is the global generating function of the identity id ∈ Dif fθ∞ and H a Sobolev small 7
−1 −1 perturbation. Explicitly ϕH (αx , y) = αx · exp−1 x (y) − αx · expx (p(η (αx ))). Bδ (x) is the neighborhood where expx is a diffeomorphism and χ a bump function with supp χ ⊂ Bδ (x). The operator σ(η) is a F IO with smooth phase function ϕH and amplitude a = 1. Moreover, σ(η) is invertible modulo smoothing operators since η is invertible and its principal symbol is a = 1, hence σ(η) ∈ (F IO0 )∗ . Furthermore, pσ(η) = η, hence σ is a local section of the exact sequence (7). Now we can use this local section σ to locally identify (F IO0 )∗ as p−1 (U ) ' σ(U ) × (ΨDO0 )∗ . This defines a local chart at the identity I ∈ (F IO0 )∗ .
• Step 4: Next we move this chart around by group structure of Dif fθ∞ (using right multiplication) and obtain an atlas for (F IO0 )∗ . • Step 5: We check that the chart transition maps are smooth and obtain (F IO0 )∗ as a smooth manifold • Step 6: We check that the group multiplication and inversion µ : (F IO0 )∗ × (F IO0 )∗ → (F IO0 )∗ , µ(A, B) = A ◦ B
(11)
ν : (F IO0 )∗ → (F IO0 )∗ , ν(A) = A−1
(12)
are ”smooth” which makes (F IO0 )∗ into a Lie group. Actually, the situation here is similar like with diffeomorphism groups. (F IO0 )∗ is an ILH-Lie group and right multiplication is smooth but left multiplication and inversion are only continuous. I will give the exact smoothness estimates in the final Theorem for open manifolds. • Step 7: Now we have the zero order operators (F IO0 )∗ as a Lie group . In order to obtain a Lie group structure on all F IO∗ we use the Laplace ∼ operator to identify (1 − ∆)m/2 : (F IO0 )∗ → (F IOm )∗ . The final result now is for the compact case: Theorem: (M. Adams, T. Ratiu, R. Schmid, [1],[2]) : The group F IO∗ (M ) of invertible Fourier integral operators on a compact manifold M is a graded ∞-dim ILH-Lie group with graded ∞-dim Lie algebra ΨDO(M ) of pseudodifferential operators on M . 8
F IO∗ (M ) is and ∞-dim principal fiber bundle over the base manifold Dif fθs (T˙ ∗ M ) of contact transformations of T˙ ∗ M with gauge group ΨDO∗ (M ) of invertible pseudodifferential operators. NON-COMPACT CASE . I like to quote my collaborator J. Eichhorn: ”There is exactly one thing that works in the non-compact case like for compact manifolds: NOTHING”. I like to illustrate what goes wrong for non compact manifolds and how we fixed it, beginning with diffeomorphism groups of open manifolds, the discussing ΨDO and F IO for non compact manifold. Details can be found in [4] and [5]. Diffeomorphisms of NON-COMPACT manifolds: Example of what goes wrong: Let M m and N n be compact manifolds, then a map f : M m → N n is of Sobolev class H s if and only if the local representatives fji : Ui ⊂ Rm → Vj ⊂ Rn are of class H s , where M ⊂ S S (Ui , φi ), N ⊂ (Vj , ψj ), fji := ψj ◦ f ◦ φ−1 i . These covers are finite if M, N are compact. This definition is invariant ⇔ s > n2 + 1. In the compact case we can define the distance by ds (f, g) := (
X
1
kfji − gji k2s ) 2 .
i,j
These definitions are meaningless if M, N are open ! The idea is to replace compactness by ”bounded geometry”. Idea: Bounded Geometry • We have control over the metric and its derivatives • We have control over the mappings and their derivatives by the metric, i.e.the maps are adapted to the bounded geometry. Definition: A Riemannian manifold (M n , g) has bounded geometry of order k, 0 ≤ k ≤ ∞, if M has a positive injectivity radius rinj (M ) and the curvature tensor R and all is derivatives up to order k are uniformly bounded; i.e the following two conditions (I) and (Bk ) are satisfied: (I) : rinj (M ) = inf rinj (x) > 0 x∈M
9
(Bk ) : |∇i R| ≤ Ci , 0 ≤ i ≤ k. Equivalent conditions are: (I) ⇔ there exists a ball around 0 in Rn which is domain of normal (geodesic) coordinates for all x ∈ M . (Bk ) ⇔ there exists a constant dk (independent of x ∈ M ) such that kgij kC k ≤ dk in any normal coordinate system ⇔ kΓm ij kC k−1 ≤ dk in any normal coordinate system Examples of manifolds with bounded geometry: Compact manifolds, Lie groups, homogeneous spaces, covering spaces of Riemannian manifolds, leaves of foliations of compact manifolds. Fact: Given an open manifold M n and k ≥ 0, then there exists a complete Riemannian metric g on M n satisfying the conditions (I) and (Bk ); i.e there is no topological obstruction for a metric with bounded geometry of any order. We now adapt all our previous constructions to the bounded geometry of the underlying manifolds. All the manifolds are assumed to have bounded geometry, i.e. satisfy conditions (I) and (Bk ). Bounded maps C ∞,r (M, N ): Consider (M, g), (N, h) open, complete Riemannian manifolds satisfying (I) and (Bk ) and f ∈ C ∞ (M, N ). Assume r ≤ k. We denote by C ∞,r (M, N ) the set of all f ∈ C ∞ (M, N ) satisfying |df |r :=
r−1 X
sup |∇i df |x < ∞.
(13)
i=0 x∈M α
∂ ν is uniformly bounded in any normal Equivalently: f ∈ C ∞,r (M, N ) ⇔ ∂x αf coordinate system; |α| ≤ r, 1 ≤ r ≤ k.
Topology on C ∞,r (M, N ): Two bounded maps f, g ∈ C ∞,r (M, N ) are said to be close iff there exists a vector field ξ along f (i.e. ξ ∈ C ∞ (f ∗ T N )) with small H p -Sobolev norm kξkp,r < ε, with 0 < ε < 21 rinj (N ), such that g(x) = expf (x) ξ(x); where kξkp,r :=
r Z X
1
|∇i t|px dvolx (g)
i=0
10
p
.
(14)
The completion of C ∞ (M, N ) with respect to this uniform structure we denote by C p,r (M, N ). Theorem: (J. Eichhorn, R. Schmid, [4]) Let 1 < p < ∞ , r ≤ k, r > np + 1. Then the completion C p,r (M, N ) is a C k+1−r -Banach manifold, and for p = 2 it is a Hilbert manifold. The bounded diffeomorphism group Dif f p,r (M ) . T Problem: We have an additional problem, namely C p,r (M ) Dif f (M ) T is not is not a group, i.e. if f ∈ C p,r (M ) Dif f (M ) 6⇒ f −1 ∈ C p,r (M ), if f is a bounded diffeomorphism 6⇒ f −1 is bounded. We need an additional assumption which implies that the bounded diffeomorphisms Dif f p,r (M ) are open in C p,r (M, M ) and each component is a C k+1−r -Banach manifold, and for p = 2 it is a Hilbert manifold. We assume that |λ|min (df ), the absolute value of the eigenvalues of the Jacobian of f , is bounded away from 0. Set Dif f p,r (M ) := {f ∈ C p,r (M, M ) | f is bijective, and |λ|min (df ) > 0}. (15) Theorem: (J. Eichhorn, R. Schmid,[4]) Let (M n , g) be an open, oriented, complete Riemannian manifold satisfying (I), (Bk ) and let r > np + 1. Then a) Dif f p,r (M ) is open in C p,r (M, M ), in particular each component is a C k+1−r -Banach manifold, and for p = 2 it is a Hilbert manifold. b) If (M, g) satisfies (B∞ ) then Dif f p,∞ (M ) = lim← Dif f p,r (M ) is an ILB - Lie group; and for p = 2 it is an ILH - Lie group. Next we discuss some important subgroups of the diffeomorphism group, the volume preserving- , symplectic- and contact diffeomorphisms. Volume preserving and symplectic diffeomorphisms Assume (M n , g) is an open manifold satisfying the conditions (I) and (Bk ), k ≥ m ≥ r > n2 + 1. Let ω be a C m -bounded closed non-degenerate qform with inf x∈M |ω|2x > 0, and consider Dif fωp,r (M ) = {f ∈ Dif f p,r (M )|f ∗ ω = ω)}. The technique to show that Dif fωp,r (M ) is a closed submanifold of Dif f p,r (M ) is to show that the map ψ : Dif f p,r (M ) → {exact q − f orms } : ψ(f ) := f ∗ (ω) 11
(16)
is a submersion,and then argue that Dif fωp,r (M ) = ψ −1 (ω) is a close submanifold. For this one needs a Sard’s theorem and a Hodge decomposition theorem which allows one to conclude that the exterior derivative operator is closed. This is not automatically given for open manifolds, so we need an additional condition. Let ∆q be the Laplace operator acting on q-forms, σe (∆q ) its essential spectrum and σe (∆q |(ker∆q )⊥ ) the essential spectrum of ∆q restricted to the orthogonal complement of its kernel, and inf σe (∆1 |(ker∆1 )⊥ ) its g.l.b. We assume the following the spectral condition inf σe (∆1 |(ker∆1 )⊥ ) > 0.
(17)
Then under this spectral condition (15) , the group of form preserving diffeomorphisms Dif fωp,r (M ) is a C k−r+1 submanifold of Dif f p,r (M ), [4]. Under the above assumptions, including (15), we have the following theorem; the technique of the proof of which is similar to the compact case, but technically more complicated. Theorem: (Eichhorn , Schmid, [4]) a) Dif fωp,∞ = lim←r Dif fωp,r is an ILH-Lie group with Lie algebra consisting of divergence free (q = n), or locally Hamiltonian (q = 2) vector fields ξ with finite Sobolev norm kξkp,r for all r. b) Dif fωp,r is an infinite dimensional Riemannian manifold, with (weak) metric Z (X, Y )x dvolx (g). (18) g(X, Y )id = M
Contact transformations of the restricted cotangent bundle T˙ ∗ M If (M n , g) is an open, oriented, complete Riemannian manifold satisfying (I), (Bk ) then the Sasaki metric on the cosphere bundle S(T ∗ M ) ⊂ T˙ ∗ M satisfies (I), (Bk−1 ). Let θ be the canonical 1-form on T ∗ M and consider Dif fθp,r = {f ∈ Dif f p,r (T˙ ∗ M )| f ∗ θ = θ }. Like in the compact case we showed in [5] that Dif fθp,s is isomorphic to the semidirect product Dif fθp,s (S(T ∗ M )) := {(ηS , h) ∈ Dif f p,s (S(T ∗ M )) >Cp,s (S(T∗ M)) | ηS∗ θS = hθS } 12
and proved the following Theorem: (Eichhorn, Schmid,[5]) Dif fθp,∞ = lim←r Dif fθp,r is an ILH-Lie group. This is the space of phase functions for the Fourier integral operators ! Pseudodifferential operators and Fourier integral operators on open manifolds Pseudodifferential (ΨDO) and Fourier integral operators (FIO) are well defined for any manifold, open or closed. But on open manifolds the spaces of these operators don’t have any reasonable structure. Moreover, many theorems for ΨDOs or FIOs on closed manifolds become wrong or don’t make any sense in the open case, e. g. certain mapping properties between Sobolev spaces of functions are wrong. The situation rapidly changes if we restrict ourselves to bounded geometry and adapt these operators to the bounded geometry. This means, roughly speaking, that the family of local symbols together with their derivatives should be uniformly bounded. For FIOs we additionally restrict ourselves to comparatively smooth Lagrangian submanifolds Λ of T˙ ∗ M × T˙ ∗ M and phase functions also adapted to the bounded geometry. Then the corresponding spaces ΨDO and F IO have similar properties as in the compact case and we can use the same ideas as before to construct Lie group structures. This is technically quite complicated and we refer for details to [5] and give here only the general ideas. Fourier integral operators are characterized by their symbols and their phase functions
F IO
Au(x) = (2π)−n
Z Z
eiϕ(x,y,ξ) a(x, ξ)u(y)dy dξ
we require roughly speaking the following: • symbols: the family of local symbols a(x, ξ) together with their derivatives should be uniformly bounded • phase functions: the phase functions ϕ(x, y, ξ) should locally generate canonical transformations in the space Dif fθp,r (T˙ ∗ M ). More precisely we define a class of uniformly bounded symbols as follows: Let B = Bε (0) ⊂ Rn be the ball which is the domain of normal (geodesic) 13
coordinates for all m ∈ M . This exists thanks our assumption (I) of bounded geometry ! Let q ∈ R and define symbol families {am }m∈M with am ∈ C ∞ (B × Rn ) , such that |∂ξα ∂xβ am (x, ξ)| ≤ Cα,β (1 + |ξ|)q−|α|
(19)
where Cα,β is independent of m ∈ M . In addition am (x, ξ) classical if P am (x, ξ) ∼ ∞ j=0 am,q−j (x, ξ) . {am }m∈M defines a family of operators am (x, Dx ) : Cc∞ (B) → C ∞ (B) , −n
am (x, Dx )u(x) := (2π)
Z Rn
Z B
am (x, ξ)u(y)dydξ.
(20)
We define the space UΨDOq (M ) of uniform pseudodifferential operators of order q as follows: A ∈ ΨDOq (M ) with Schwartz kernel KA belongs to UΨDOq (M ) iff (i) ex. const. CA > 0, s.t. KA (x, y) = 0 for d(x, y) > CA (ii) KA is smooth outside the diagonal of M × M (iii) for any δ > 0, i, j ex.const Cδ,i,j > 0 s.t. |∇ix ∇jy KA (x, y)| ≤ Cδ,i,j , d(x, y) > δ (iv) define Am := exp∗m ◦ A ◦ (exp∗m )−1 : Cc∞ (B) → C ∞ (B) has the form Am = am (x, Dx ) + Rm . (Rm smoothing operator). Note that −k ≤ q ⇒ UΨDO−k−1 ⊂ UΨDOq and denote UΨDOq,k := UΨDOq /UΨDO−k−1 and s UΨDOq,k its completion in the H s Sobolev norm. Now we define a uniform family of nondegenerate phase functions {ϕm }m∈M by requiring that each ϕm (x, y, ξ) is locally generating a canonical transfor∗ mation ηm : T˙ ∗ M → T˙ ∗ M : ηm θ = θ , with ηm ∈ Dif fθp,r (T˙ ∗ M ). With this we define the class of uniform Fourier integral operators of order q, denoted by UF IOq as follows: A ∈ UF IOq iff A : Cc∞ (M ) → C ∞ (M ) and locally Am := exp∗m ◦ A ◦ (exp∗m )−1 : Cc∞ (B) → C ∞ (B) is of the form −n
Am u(x) = (2π)
Z
Z
Rn
B
eiϕm (x,y,ξ) am (x, y, ξ)u(y)dydξ
(21)
where am is a uniformly bounded classical symbol of order q and ϕm a uniform phase function. Let UF IOq,k := UF IOq /UF IO−k−1 . We get the exact sequence of groups j
p
I → (UΨDO0,k )∗ ,→ (UF IO0,k )∗ → Dif fθp,r → id 14
(22)
Now we follow the same ideas as in the compact case: Performing Steps 1,2...7 as before to construct ILH Lie groups structures on these spaces. The openness of the underlying manifold always requires additional considerations and additional estimates, e.g. the ILH-Lie algebras are quite different from the ones in the compact case. We refer to [5] for details and state the main result: Main Theorem: (Eichhorn, Schmid, [5]) Assume (M n , g) ia an open Riemannian manifold satisfying the conditions (I) and (B∞ ) of bounded geometry and the condition inf σe (41 (S(T ∗ M )), gS |ker41 )⊥ ) > 0 . Then for any k ∈ Z+ 2,∞ ˙ ∗ 2,r ˙ ∗ 1. Dif fθ,0 (T M ) = lim Dif fθ,0 (T M ) , r ≥ n + 1, is an ILH Lie group. ←s s 2. (UΨDO0,k )∗ = lim(UΨDO0,k )∗ , s ≥ n + 1, is an ILH Lie group and each ←s s (UΨDOo,k )∗ is a smooth Hilbert Lie group. t )∗ , t > max{2n, n + 2(k + 1)}} is an ILH Lie 3. (UF IOo,k )∗ = lim(UF IOo,k ←t group with the following properties. t a. (UF IO0,k )∗ is a C t Hilbert manifold modeled on C 2,t+1 (S(T ∗ M )) × t−2(k+1) )∗ . (UΨDO0,k t )∗ is modeled by C 2,t+1 (S) ⊕ C 2,k+t−(2k+1) (S) ⊕ Each component of (UF IO0,k · · · ⊕ C 2,t−2(k+1) (S). t+1 t )∗ is a C t map b. The inclusion (UF IO0,k )∗ ,→ (UF IO0,k l c. The group multiplication µ is a C map , l = min{r, t} t+r t t µ : (UF IO0,k )∗ × (UF IO0,k )∗ −→ (UF IO0,k )∗ , µ(A, B) = A ◦ B
d. The inversion ν is a C l map, l = min{r, t} t+r t )∗ , ν(A) = A−1 ν : (UF IO0,k )∗ −→ (UF IO0,k
t e. Right multiplication RA by A ∈ (UF IO0,k )∗ is a C t map t t )∗ , RA (B) = B ◦ A. RA : (UF IO0,k )∗ −→ (UF IO0,k
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APPLICATIONS We discuss tree applications. The classical Euler’s equations for incompressible fluids, the topological Euler’s equations and the KdV equation. 1. Euler’s equations 1) Classical Euler equations for an incompressible, homogeneous fluid without viscosity (
Ecl
∂u ∂t
+ ∇u(t) u(t) = grad p div u(t) = 0
where u = u(x, t) is a time dependent C 1 vector field on (M n , g), ∇ = ∇g , div = divdvolx (g) . Additionally, we assume u(t) ∈ C r (T M ) for all t which means that the fluid moves very slowly at infinity , r > n2 + 1. Then u(t) defines a 1-parameter family of diffeomorphisms ft defined by dfs |s=t = u(t) ◦ ft . ds Theorem: (Eichhorn, Schmid) Assume (M n , g) open with (I) and (Bk ), k − 2 ≥ r > n2 + 1. Then u(t) satisfies the classical Euler equations (Ecl ) iff {ft }t is a geodesic in Dif fµ∞,r (M ). M compact: D.G. Ebin and J.E. Marsden (1970). 2) Topological Euler equations µ = fixed volume form on (M, g) u = u(x, t) is a time dependent C 1 vector field on (M n , g) ∇ = ∇g the Riemannian covariant derivative but now div = divµ , defined by LX µ = (divµ X)µ. (
(Etop )
∂u ∂t
+ ∇u(t) u(t) = grad p divµ u(t) = 0
Theorem: (Eichhorn, Schmid) Assume (M n , g) open with (I) and (Bk ), k − 2 ≥ r > n2 + 1. Then u(t) satisfies the topological Euler equations (Etop ) iff {ft }t is a geodesic in Dif fµ∞,r (M ).
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2. KdV equation and the group of Fourier integral operators Korteweg deVries (KdV) equation u = 6uux − uxxx R δF t δG Poisson bracket {F, G}(u)R = δu ∂x δu dx Hamiltonian H(u) = (u3 + 12 u2x )dx Hamilton’s equations ut = {u, H} ⇐⇒ u satisfies KdV Gardner, Kruskal 1971 Theorem: (M.Adams, J.Eichhorn, T.Ratiu, R.Schmid) A: The KdV equation is a Hamiltonian system with respect to the Lie-Poisson bracket on the coadjoint orbit of the Lie group of invertible Fourier integral operators G = F IO∗ through the Schr¨odinger operator. B: The Kostant-Symes theorem applied to a splitting of the Lie algebra of F IO∗ , the space of pseudodifferential operators g = ΨDO gives the complete integrability of KdV, i.e. the Gelfand-Dikii family of commuting integrals, including H. Consider M = S 1 the unit circle. Then each pseudodifferential operator P P ∈ ΨDOm (S 1 )has total symbol of the form p(x, ξ) = −∞
