Paradifferential Calculus in Gevrey Classes - Project Euclid

J. Math. Kyoto Univ. (JMKYAZ) 41-1 (2001), 1–31

Paradifferential Calculus in Gevrey Classes By Chen Hua and Luigi Rodino

Abstract

We present a paradifferential calculus adapted to the study of nonlinear partial differential equations in Gevrey classes. We give an application concerning Gevrey microregularity of the solutions of fully nonlinear equations at elliptic points.

1.

Introduction

Bony [4], 1981, presented a version of the pseudo-differential calculus, the so-called paradifferential calculus, adapted to the study of the nonlinear equations (1.1)

F [u] = F (x, u, · · · , ∂ α u, · · · )|α|≤m = 0.

The basic idea was to write (1.2)

F [u] = TF 0 (u) u + r,

where TF 0 (u) is the paradifferential operator having as symbol the linearization F 0 of F at u, and r is a smooth error. Precisely, if u is assumed of Sobolev class H s+m , s > n/2, then r ∈ H 2s−n/2 . Through (1.2) one is reduced to the study of the paradifferential equation TF 0 (u) u ∈ H 2s−n/2 and obtains linear-type results of existence, regularity and propagation. Assume now F is analytic in the respective variables, and let u be of Gevrey class Gσ , i.e. locally (1.3)

|∂ α u(x)| ≤ C |α|+1 (α!)σ .

Naively, one could try to reproduce for the Gevrey scale Gσ , 1 < σ < ∞, the results of Bony for the scale H s , n/2 < s < ∞. It is easily seen, however, that the error r in (1.2) turns out to be of class Gσ , with apparent no gain of regularity. This corresponds to the known impossibility of obtaining linear-type results of propagation for (1.1) in the analytic-Gevrey category, but for very Communicated by Prof. K. Ueno, January 26, 1999 Revised October 3, 2000

2

Chen Hua and Luigi Rodino

special equations, cf. Alinhac and Métivier [1], Godin [8], Chen and Rodino [5], [6] and Sasaki [17]. Here we propose a different approach, developing some preliminary results s of Chen and Rodino [5]. Namely, we refer to the Sobolev-Gevrey spaces Hτ,σ , 1 < σ < ∞, τ > 0, s ∈ IR, defined by (1.4)

1/σ s =k exp[τ hDi k u kHτ,σ ]u kH s < ∞.

Spaces of this type have been already studied, for example, in Kajitani and Nishitani [13], Kajitani and Wakabayashi [14] and Taniguchi [18]. s s , n/2 < . For fixed σ and τ , the scale Hτ,σ Locally we have Gσ = ∪τ,s Hτ,σ s < ∞, is indeed appropriate for the paradifferential calculus, the error r in 2s−n/2 s (1.2) belonging to Hτ,σ , if u ∈ Hτ,σ . As an application, we may extend to the fully nonlinear case the result of micro-elliptic Gevrey regularity proved in [5] for the semilinear equations. s 2s−λ Namely, from u ∈ Hτ,σ we deduce u ∈ Hτ,σ , for some fixed constant λ, microlocally in (1.1) at any elliptic point. Though very weak, this result is nearly the best possible; in fact, starting from a solution u ∈ Gσ , there is no 0 hope in general of reaching Gσ regularity, where 1 ≤ σ 0 < σ, as in the linear case, see the counter-examples in [5] and [6]. Other possible applications in the nonlinear setting, which we leave to the future, concern Gevrey propagation, cf. Bony [4] and Hörmander [11] for the H s category, existence of Gevrey Riemannian embeddings, cf. Hörmander [10] and Chen and Rodino [7], existence of Gevrey solutions for equations with multiple characteristics, cf. Gramchev and Rodino [9], and weakly hyperbolic Cauchy problems, cf. Mizohata [15] and Kajitani [12]. Finally, we observe that part of the arguments in the following may keep valid for other weighted Sobolev spaces; essentially, one can replace the operator exp[τ hDi1/σ ] in (1.4) with a more general operator Φ(D), provided Φ(ξ + η) ≤ CΦ(ξ)Φ(η). The paper is organized as follows: in Section 2, we recall the results of s spaces; in Section 3, we [5] about Littlewood-Paley decomposition and Hτ,σ treat some classes of Gevrey pseudo-differential operators; in Sections 4 and 5, we study paraproducts and paradifferential operators, respectively, in the s Hτ,σ frame; in Section 6, we present the above-mentioned application to microellipticity. 2.

Gevrey-Sobolev spaces and non-linear operations Let σ > 1, τ, s ∈ IR; we introduce Gevrey-Sobolev spaces as follows: s 0 Hτ,σ (IRn ) = {u ∈ S−τ,σ (IRn ), exp[τ hDi1/σ ]u ∈ H s (IRn )},

0 where hDi = (1 − ∆)1/2 , the space Sτ,σ is defined as the dual of Sτ,σ , which in turn for τ ≥ 0 is defined by inverse Fourier transform from

Sbτ,σ = {v(ξ) ∈ C ∞ (IRn ) | exp[τ hξi1/σ ]v(ξ) ∈ S(IRn )};

Paradifferential Calculus in Gevrey Classes

3

for τ < 0, the space Sτ,σ is defined by transposition of inverse Fourier transform s from Sbτ,σ (cf. [14]). We shall also write Hτ,σ for short. The infinite order 1/σ pseudo-differential operator exp[τ hDi ] is defined by Fourier transform as usual; see for example Rodino [16]. s We define the norms in Hτ,σ by 1/σ s = k exp[τ hDi k u kHτ,σ ]ukH s . s Hτ,σ is a Hilbert space with inner product s hu, viHτ,σ = hexp[τ hDi1/σ ]u, exp[τ hDi1/σ ]viH s .

Taking K > 1 a constant, for p ∈ Z+ , we denote: Cp = {ξ ∈ IRn , K −1 2p ≤ |ξ| ≤ K2p+1 }, C−1 = B(0, K) = {ξ ∈ IRn , |ξ| ≤ K}. n Thus {Cp }∞ p=−1 is a circular cover of IRξ . From Bony [4], we have the following result:

Lemma 2.1. There exists N1 ∈ N, depending only on K, such that for any Cp the number of q, such that Cq ∩ Cp 6= ∅, is at most N1 . We have the following dyadic partition of unity, see again Bony [4]: Lemma 2.2. There exist ϕ, ψ ∈ C0∞ (IRn ), supp ψ ⊂ C−1 , supp ϕ ⊂ C0 , such that for any ξ ∈ IRn , and any l ∈ N, we have (2.1)

ψ(ξ) +

∞ X

ϕ(2−p ξ) = 1,

p=0

(2.2)

ψ(ξ) +

l−1 X

ϕ(2−p ξ) = ψ(2−l ξ).

p=0

The Littlewood-Paley decomposition (or say dyadic decomposition) s {up }∞ p=−1 for a function u ∈ Hτ,σ will be defined as follows: u−1 (x) = ψ(D)u(x), up (x) = ϕ(2−p D)u(x) for p ≥ 0. P∞ 0 It is easy to prove that the series u = p=−1 up is convergent in the S−τ,σ topology. In fact we can use the dyadic decomposition to characterize the Gevrey-Sobolev spaces. (2.3)

Theorem 2.1. Let s > 0, σ > 1 and τ ∈ IR; then the following conditions are equivalent: s (a) u ∈ Hτ,σ (IRn ); P∞ (b) u = p=−1 up , where up ∈ C ∞ and supp u ˆp ⊂ Cp , satisfying kup kL2τ,σ 0 ≤ cp 2−ps with {cp } ∈ `2 , L2τ,σ = Hτ,σ ;

4


P∞ (c) u = p=−1 up , where up ∈ C ∞ and supp u ˆp ⊂ B(0, K1 2p ) for some K1 > 0, satisfying kup kL2τ,σ ≤ cp 2−ps , {cp } ∈ `2 ; P∞ ∞ (d) u = and for any α ∈ Zn+ , we have p=−1 up , where up ∈ C kDα up kL2τ,σ ≤ cpα 2−ps+p|α| and {cpα }p ∈ `2 . It is obvious that the proof of Theorem 2.1, for τ > 0, may be deduced directly from the proof of [5, Theorem 1.1], similarly we can prove the case for τ ≤ 0 as well. Remark 2.1.

In (d) it is actually sufficient to argue for |α| ≤ s + 1.

s can be estimated in (b) and (c) by Remark 2.2. The P norm kukHτ,σ k(cp )k`2 and in (d) by |α|≤s+1 k(cp,α )p k`2 . The equivalence between (a) and (b) keeps valid for s ∈ IR.

Remark 2.3. Let us recall, see for example Rodino [16], that every ϕ ∈ Gσ0 (IRn ) = Gσ (IRn ) ∩ C0∞ (IRn ), σ > 1, satisfies for suitable positive constants C and ε: |ϕ(ξ)| ˆ ≤ C exp[−ε|ξ|1/σ ].

(2.4) 0

s (IRn ) for any σ 0 < σ, s ∈ IR and τ > 0, with It follows that Gσ0 (IRn ) ⊂ Hτ,σ s (IRn ) for all s, if strict inclusion; moreover if ϕ ∈ Gσ0 (IRn ), then ϕ ∈ Hτ,σ τ > 0 is sufficiently small. In the opposite direction, it is easy to see that s Hτ,σ (IRn ) ⊂ Gσ (IRn ) for all τ > 0, s ∈ IR; cf. the proof of Theorem 1.6.1, (iii) in [16]. 0 s s . Thus and ϕ ∈ Gσ0 (IRn ), 1 < σ 0 < σ, then ϕu ∈ Hτ,σ Observe if u ∈ Hτ,σ we can define Gevrey locally Sobolev spaces as follows (cf. [5]): s Definition 2.1. We set Hτ,σ,loc to be the space of all Gevrey ultra0 0 n distributions u ∈ Dσ0 (IR ) such that for every ϕ ∈ Gσ0 (IRn ) with 1 < σ 0 < σ, s we have ϕu ∈ Hτ,σ .

We also define, for some open subset Ω ⊂ IRn , that 0

s Definition 2.2. We say u ∈ Hτ,σ,loc (Ω), if for all ϕ ∈ Gσ0 (Ω), 1 < σ 0 < s s σ, we have ϕu ∈ Hτ,σ . We say u ∈ Hτ,σ (x0 ) for x0 ∈ IRn if there exists a s neighborhood Vx0 of x0 , such that u ∈ Hτ,σ,loc (Vx0 ). s Observe that ∪s∈IR,τ >0 Hτ,σ (x0 ) = Gσ (x0 ), the space of all the functions 0 s u which are of class Gσ in a neighborhood of x0 ; moreover Gσ (x0 ) ⊂ Hτ,σ (x0 ) 0 with strict inclusion for all s ∈ IR, τ > 0 and 1 < σ < σ. Let (x0 , ξ 0 ) ∈ T ∗ IRn \{0}; we introduce Gevrey microlocally (i.e. near (x0 , ξ 0 )) Sobolev spaces as follows: s Definition 2.3. We write u ∈ Hτ,σ (x0 , ξ 0 ) if there exists Vx0 and a 0 0 n conic neighborhood Γ0 of ξ in IR \{0}, such that for all ϕ ∈ Gσ0 (Vx0 ), 1 < σ 0 < σ, and every ψ ∈ C ∞ (IRnξ ), 0-order homogeneous in ξ for large |ξ| with s . supp ψ ⊂ Γ0 , we have ψ(D)(ϕu) ∈ Hτ,σ


5

+∞ −∞ s s Next Hτ,σ and Hτ,σ will be defined by ∩s Hτ,σ and ∪s Hτ,σ respectively. +∞ −∞ Similarly we can define Hτ,σ,loc (Ω) and Hτ,σ,loc (Ω). To be definite, we recall here the notion of Gevrey wave front set and related remarks used in the sequel.

Definition 2.4. For u ∈ Dσ0 (IRn ) we write (x0 , ξ 0 ) 6∈ WF σ u if there exσ n ist ϕ ∈ G0 (IR ), with ϕ = 1 in a neighborhood of x0 , and a conic neighborhood Γ0 of ξ 0 such that for positive constants C, ε: (2.5)

|(ϕu)ˆ(ξ)| ≤ C exp[−ε|ξ|1/σ ], ξ ∈ Γ0 .

Remark 2.4. Equivalently (cf. Rodino [16, Lemma 1.7.3]), we may say that (x0 , ξ 0 ) 6∈ WF σ u if there exist Vx0 and Γ0 such that for all ϕ ∈ Gσ0 (Vx0 ) and every ψ ∈ C ∞ (IRnξ ), 0-order homogeneous in ξ for large |ξ| with supp ψ ⊂ Γ0 , we have for some C, ε > 0: (2.6)

|(ψ(ξ)(ϕu))ˆ(ξ)| ≤ C exp[−ε|ξ|1/σ ].

s It is then clear that (x0 , ξ 0 ) 6∈ WF σ0 u for 1 < σ 0 < σ implies u ∈ Hτ,σ (x0 , ξ 0 ) for all τ > 0, s ∈ IR.

We have the following Hausdorff-Young inequality: Theorem 2.2.

Let u ∈ L1 , v ∈ L2τ,σ , τ ∈ IR, σ > 1, then ku ∗ vkL2τ,σ ≤ kukL1 · kvkL2τ,σ .

Moreover we can easily extend the results in [5, Theorems 2.1 through 2.3] to the case of τ ≤ 0, i.e. we have the following results: s Theorem 2.3. We have u ∈ Hτ,σ (x0 , ξ 0 ) (τ , s ∈ IR, σ > 1) if and only if there exist Vx0 and a decomposition

u = u1 + u2 , for x ∈ Vx0 , s where u1 ∈ Hτ,σ (IRn ) and (x0 , ξ 0 ) 6∈ WF σ0 (u2 ) for 1 < σ 0 < σ.

Theorem 2.4. Let s0 > s, τ ∈ IR and σ > 1, the following two conditions are equivalent: s s0 (a) u ∈ Hτ,σ (x0 ) ∩ Hτ,σ (x0 , ξ 0 ). 0 (b) There exists ϕ1 ∈ Gσ0 (IRn ), 1 < σ 0 < σ, with ϕ1 = 1 in a neighborhood Vx0 of P x0 , and there exists a conic neighborhood Γ0 of ξ 0 such that 0 ∞ ϕ1 u = u−1 + p=0 (u0p + u00p ), where u−1 ∈ Gσ (IRn ), ku0p kL2τ,σ ≤ c0p 2−ps , 0

C 00 00 −ps supp u ˆ0p ⊂ Cp ∩ΓC , supp u ˆ00p ⊂ Cp , 0 (Γ0 is complement of Γ0 ), kup kL2τ,σ ≤ cp 2 0 00 2 {cp }, {cp } ∈ ` .

Theorem 2.5. Let u ∈ L2τ,σ (or u ∈ L2|τ |,σ ), τ ∈ IR, σ > 1, and v ∈ s s 2 H|τ |,σ (or v ∈ Hτ,σ ), with s > n/2. Then uv ∈ Lτ,σ and for some C > 0 : s kuvkL2τ,σ ≤ CkvkH|τ kukL2τ,σ |,σ

(or

s kuvkL2τ,σ ≤ CkvkHτ,σ kukL2|τ |,σ ).

6


It is important for us to consider when a function space would become an algebra, which in particular is useful to study nonlinear partial differential equations. Here for Gevrey-Sobolev spaces we have (similar to [5]): Theorem 2.6. Let s > n/2 and τ ≥ 0, then s (a) Hτ,σ is an algebra, and there exists C > 0 such that for all u, v ∈ s Hτ,σ , s > n/2: s s s . kuvkHτ,σ ≤ CkukHτ,σ kvkHτ,σ (b) algebra. 3.

0

s s (x0 ) ∩ Hτ,σ (x0 , ξ 0 ) is an Furthermore, if s0 < 2s − n/2, then Hτ,σ

Some classes of Gevrey pseudo-differential operators We first define some classes of Gevrey symbols.

Definition 3.1. Let m ∈ IR, τ ≥ 0, σ > 1 and ε > 0. We denote by m,ε , the class of all the symbols p(x, ξ) ∈ C ∞ (IRn × IRn ) such that Sτ,σ (3.1)

kDξβ p(·, ξ)kH n/2+ε ≤ cβ hξim−|β| , τ,σ

for constants cβ independent of ξ ∈ IRn . Denoting by pˆ(η, ξ) the partial Fourier transform of p(x, ξ) with respect to the first variables, we can re-write (3.1) as (3.2)

k exp[τ hDx i1/σ ]Dξβ p(x, ξ)kH n/2+ε (IRnx ) = k exp[τ hηi1/σ ]hηin/2+ε Dξβ pˆ(η, ξ)kL2 (IRnη ) ≤ cβ hξim−|β| .

m,ε In our setting the classes Sτ,σ play the role of symbols with “limited smoothm,ε and Σm,ε ness”; they will contain the subclasses lτ,σ τ,σ of the Gevrey paradifferential symbols, see the next sections for precise definitions. Here let us fix attention on the corresponding class of “smooth” symbols. m Definition 3.2. Let m ∈ IR, τ ≥ 0, σ > 1. We denote by Sτ,σ the class m,ε m given by ∩ε>0 Sτ,σ ; i.e. p(x, ξ) ∈ Sτ,σ if (3.1), or (3.2), is valid for every ε > 0. m Equivalent definitions for Sτ,σ are obtained by imposing for every α, β:

(3.3)

s kDxα Dξβ p(·, ξ)kHτ,σ ≤ cαβ hξim−|β| ,

for some fixed s ∈ IR; in particular for s = 0 (3.4)

kDxα Dξβ p(·, ξ)kL2τ,σ ≤ cαβ hξim−|β| ,

−∞ m with suitable constants cαβ . Let us write Sτ,σ = ∩m Sτ,σ . 0

s Hτ,σ

Example 3.1. Let φ ∈ Gσ0 (IRn ), 1 < σ 0 < σ. Then φ(x) belongs to 0 for every τ > 0, s ∈ IR, and it can be regarded as symbol in Sτ,σ .


7

Example 3.2. Let a(ξ) ∈ C ∞ (IRn ) satisfy the estimates |Dξβ a(ξ)| ≤ m cβ hξim−|β| . Then φ(x)a(ξ), with φ(x) as in Example 3.1, belongs to Sτ,σ . 0

0

m m m+m If p ∈ Sτ,σ , q ∈ Sτ,σ , then pq ∈ Sτ,σ , as we have from (3.1), the Leibniz n/2+ε

m−|β|

m rule and the algebra property of Hτ,σ . If p ∈ Sτ,σ , then Dxα Dξβ p ∈ Sτ,σ . mj Let pj ∈ Sτ,σ , j =P 1, 2, · · · , mj → −∞ with mj+1 ≤ mj for all j, and let ∞ m1 p ∈ Sτ,σ ; we write p ∼ j=1 pj if for all integers N ≥ 2 we have

p−

X

mN pj ∈ Sτ,σ .

1≤j 1. We denote by Sτ,σ,cl P∞ m the subclass of all p(x, ξ) ∈ Sτ,σ such that p(x, ξ) ∼ j=0 pm−j (x, ξ) where m−j pm−j (x, ξ) ∈ Sτ,σ is positively homogeneous with respect to ξ of degree m − j, for large |ξ|.

For later reference we also introduce more general classes of ρ, δ-type. Definition 3.4. Let m ∈ IR, τ ≥ 0, σ > 1, 0 ≤ ρ ≤ 1, 0 ≤ δ ≤ 1. We m denote by Sτ,σ,ρ,δ the class of all p(x, ξ) ∈ C ∞ (IRn × IRn ) satisfying for every α, β kDxα Dξβ p(·, ξ)kL2τ,σ ≤ cαβ hξim−ρ|β|+δ|α| .

(3.5)

m m = Sτ,σ . In particular we have Sτ,σ,1,0

Consider now pseudo-differential operators Z P u(x) = p(x, D)u(x) = (2π)−n eixξ p(x, ξ)ˆ u(ξ)dξ, with symbol in the preceding classes. For simplicity we shall limit ourselves m to the main properties of the operators with symbols in Sτ,σ ; variants and m generalizations to Sτ,σ,ρ,δ are left to the reader. Proofs will follow closely the calculus for pseudo-differential operators with limited Sobolev smoothness, see for example Beals [2], Beals and Reed [3] and Taylor [19]. In particular the following lemma, taken from Beals and Reed [3], will be very useful in our context. Lemma 3.1. Suppose that Z Z C 2 = sup |g(λ, ξ)|2 dλ < ∞ and K 2 = sup |G(ξ, η)|2 dξ < ∞. ξ∈IRn

η∈IRn

8


For h ∈ L2 define

Z

Ah(η) =

G(ξ, η)g(η − ξ, ξ)h(ξ)dξ.

Then kAhkL2 ≤ CKkhkL2 . The proof of Lemma 3.1 is elementary, by writing kAhkL2 = supkf k≤1 | f (η)Ah(η)dη|, interchanging integrals and using Schwarz inequality. R

m Theorem 3.1. If p(x, ξ) ∈ Sτ,σ (m ∈ IR, τ ≥ 0, σ > 1), then P : s−m s s−m s Hτ,σ → Hτ,σ , P : H−τ,σ → H−τ,σ continuously for every s ∈ IR. −∞ −∞ +∞ +∞ . In particular we have → Hτ,σ , P : Hτ,σ → Hτ,σ It follows P : Hτ,σ −∞ , then P is regularizing, in the sense that that if p(x, ξ) ∼ 0, i.e. p(x, ξ) ∈ Sτ,σ −∞ +∞ P : Hτ,σ → Hτ,σ . s s−m continuously. We first write Proof. Let us prove P : Hτ,σ → Hτ,σ Z Pû(η) = (2π)−n pˆ(η − ξ, ξ)ˆ u(ξ)dξ,

where as before pˆ is the partial Fourier transform of p with respect to the x-variables. We then have to estimate the L2 -norm of (3.6) exp[τ hηi1/σ ]hηis−m (Pû)(η) Z −n = (2π) H(ξ, η)hηis−m hξi−s exp[τ hη − ξi1/σ ]ˆ p(η − ξ, ξ)ˆ v (ξ)dξ, where

v = exp[τ hDi1/σ ]hDis u,

s so that kukHτ,σ = kvkL2 , and we have set

H(ξ, η) = exp[τ hηi1/σ − τ hξi1/σ − τ hη − ξi1/σ ]. Note that H(ξ, η) ≤ 1. We apply Lemma 3.1 by taking there g(λ, ξ) = exp[τ hλi1/σ ]hλiN pˆ(λ, ξ)hξi−m , which satisfies for every fixed N

Z

2

|g(λ, ξ)|2 dλ < ∞,

C = sup ξ∈IRn

in view of Definition 3.2. We set also G(ξ, η) = H(ξ, η)hηis−m hξim−s hη − ξi−N , for which

Z 2

K = sup

|G(ξ, η)|2 dξ < ∞

η∈IRn

if N has been chosen sufficiently large. Therefore by Lemma 3.1, the L2 -norm of (3.6) is estimated by CKkvkL2 , and this gives the conclusion. Similarly we s−m s prove P : H−τ,σ → H−τ,σ .


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Remark 3.1. Concerning symbols with limited Gevrey-Sobolev m,ε smoothness, p(x, ξ) ∈ Sτ,σ , the corresponding operators act with continuity 0 s s 0 from Hτ,σ to Hτ,σ , s ≤ s − m, provided s is sufficiently large and s0 sufficiently small, depending on ε and m. m Theorem 3.2. (1) Let p(x, ξ) ∈ Sτ,σ and consider the corresponding pseudo-differential operator P . Define the L2 -adjoint P ∗ by

hP ∗ u, viL2 = hu, P viL2 ,

−s s+m u ∈ Hτ,σ , v ∈ H−τ,σ .

m Then P ∗ is a pseudo-differential operator with symbol p∗ (x, ξ) ∈ Sτ,σ , having asymptotic expansion X (3.7) p∗ (x, ξ) ∼ (α!)−1 ∂ξα Dxα p¯(x, ξ). α m Sτ,σj ,

(2) Let pj ∈ j = 1, 2, and consider the corresponding pseudodifferential operators Pj , j = 1, 2. Then P1 P2 is a pseudo-differential operm1 +m2 ator with symbol p(x, ξ) ∈ Sτ,σ , having asymptotic expansion (3.8)

p ∼ p1 #p2 =

X

(α!)−1 ∂ξα p1 (x, ξ)Dxα p2 (x, ξ).

α

Proof. We prove (2) and leave (1) to the reader. By standard computations we may express the symbol of P1 P2 in the form Z −n (3.9) p(x, ζ) = (2π) exp[−i(y − x)(ξ − ζ)]p1 (x, ξ)p2 (y, ζ)dydξ Z = (2π)−n exp[ixξ]p1 (x, ζ + ξ)ˆ p2 (ξ, ζ)dξ, where pˆ2 is the Fourier transform of p2 with respect to the x-variables. m1 +m2 Let us first show that p(x, ζ) ∈ Sτ,σ . To this end, we compute pˆ(η, ζ), Fourier transform of p(x, ζ) with respect to the x-variables, and obtain Z −n pˆ(η, ζ) = (2π) pˆ1 (η − ξ, ζ + ξ)ˆ p2 (ξ, ζ)dξ. We have to estimate for any s ∈ IR the L2 -norm with respect to η of hζi−m1 −m2 +|β| exp[hηi1/σ ]hηis Dζβ pˆ(η, ζ), uniformly in the parameter ζ. Let us limit ourselves to treat the case β = 0, the generalization to arbitrary β being trivial by Leibniz rule. We have (3.10) hζi−m1 −m2 exp[τ hηi1/σ ]hηis pˆ(η, ζ) Z −n H(ξ, η)hηis hζi−m1 −m2 pe1 (η − ξ, ζ + ξ)e p2 (ξ, ζ)dξ, = (2π)

10


where H(ξ, η) = exp[τ hηi1/σ − τ hξi1/σ − τ hη − ξi1/σ ] ≤ 1 and pe1 (η − ξ, ζ + ξ) = exp[τ hη − ξi1/σ ]ˆ p1 (η − ξ, ζ + ξ), 1/σ pe2 (ξ, ζ) = exp[τ hξi ]ˆ p2 (ξ, ζ). We shall apply again Lemma 3.1, all the terms there depending on the parameter ζ with uniformly bounded norms. Namely, we set h(ξ, ζ) = pe2 (ξ, ζ)hζi−m2 hξiL with L2 -norm with respect to ξ bounded uniformly with respect to ζ, for any L; moreover g(λ, ξ, ζ) = pe1 (λ, ζ + ξ)hζ + ξi−m1 λM with L2 -norm with respect to λ bounded uniformly with respect to ζ and ξ, for any M . Finally, we take G(ξ, η, ζ) = H(ξ, η)hηis hζi−m1 hξi−L hζ + ξim1 hη − ξi−M , for which

Z |G(ξ, η, ζ)|2 dξ < ∞

sup η,ζ

if L and M have been chosen sufficiently large. From Lemma 3.1 we therefore deduce that the L2 -norm with respect to η of (3.10) is bounded, uniformly with respect to ζ. We pass now to prove the asymptotic formula in (2). As standard in the pseudo-differential calculus, after Taylor expanding p1 (x, ζ + ξ) in (3.9) with respect to ξ, we are reduced to consider the remainder rN (x, ζ) =

X NZ 1 rN γ (x, ζ, t)(1 − t)N −1 dt, γ! 0

|γ|=N

where Z rN γ (x, ζ, t) = (2π)−n

exp[ixξ]∂ξγ p1 (x, ζ + tξ)ξ γ pˆ2 (ξ, ζ)dξ.

m1 +m2 −N , N = |γ|, with uniform bounds with We have to prove that rN γ ∈ Sτ,σ respect to the parameter t, 0 ≤ t ≤ 1. Arguing as before, we are led to consider Z rˆN γ (η, ζ, t) = (2π)−n ∂ξγ pˆ1 (η − ξ, ζ + tξ)ξ γ pˆ2 (ξ, ζ)dξ.

Repeating the preceding arguments, and in particular applying Lemma 3.1 with ζ and t as parameters, we get easily the conclusion. −∞ s m Corollary 3.1. If u ∈ Hτ,σ , u ∈ Hτ,σ (x0 ) and P has symbol in Sτ,σ , s−m then P u ∈ Hτ,σ (x0 ).


11

0

Proof. Take φ ∈ Gσ0 (IRn ), 1 < σ 0 < σ, with φ(x) = 1 in a neighborhood 0 s V of x0 , such that φu ∈ Hτ,σ . Take then any φ0 ∈ Gσ0 (IRn ) with supp φ0 ⊂ V . Writing φ0 P u = φ0 P φu + φ0 P (1 − φ)u, applying Theorem 3.1 to the first term in the right hand side and (3.8) in s−m s−m Theorem 3.2 to the second term, we get φ0 P u ∈ Hτ,σ , hence P u ∈ Hτ,σ (x0 ). −∞ s Corollary 3.2. If u ∈ Hτ,σ , u ∈ Hτ,σ (x0 , ξ 0 ) and P has symbol in 0 s−m then P u ∈ Hτ,σ (x0 , ξ ).

m , Sτ,σ

Proof. Let ψ(ξ), ψ 0 (ξ) ∈ C ∞ (IRn ) be 0-order homogeneous for large |ξ| with ψ(ξ) = 1 in a conic neighborhood Γ of ξ 0 and supp ψ 0 ⊂ Γ. Let φ, φ0 be s as in the preceding proof, such that ψ(D)(φu) ∈ Hτ,σ . The conclusion is easily obtained by writing ψ 0 (D)(φ0 P u) = ψ 0 (D)φ0 P (ψ(D)φu) + ψ 0 (D)φ0 P (1 − ψ(D)φ)u and applying (3.8) to the second term in the right hand side. m Given Ω open subset of IRn , the class of symbols Sτ,σ (Ω) is the set of all 0 ∞ n m p(x, ξ) ∈ C (Ω × IR ) such that φ(x)p(x, ξ) ∈ Sτ,σ for every φ ∈ Gσ0 (Ω), 1 < m,ε m m σ 0 < σ. Similarly we define Sτ,σ (Ω), Sτ,σ,cl (Ω) and Sτ,σ,ρ,δ (Ω). The preceding Theorems 3.1 and 3.2 have obvious variants for the corresponding pseudodifferential operators. P∞ m Let p(x, ξ) ∼ j=0 pm−j (x, ξ) in Sτ,σ,cl (Ω) be elliptic, i.e. for every K ⊂⊂ Ω we have |pm (x, ξ)| ≥ cK |ξ|m , x ∈ K, ξ ∈ IRn , −m for a suitable positive constant cK . Then q−m (x, ξ) = (pm (x, ξ))−1 ∈ Sτ,σ (Ω) for large ξ and we may recursively construct as standard q(x, ξ) ∼ P∞ −m q (x, ξ) in S (Ω), such that q#p = 1, p#q = 1. From (2) in j=0 −m−j τ,σ,cl Theorem 3.2 we therefore obtain: m Theorem 3.3. Let p(x, ξ) ∈ Sτ,σ,cl (Ω) be elliptic in Ω. Assume P = +∞ p(x, D) is properly supported, i.e. it is well defined as a map P : Hτ,σ,loc (Ω) → +∞ −∞ −∞ Hτ,σ,loc (Ω), Hτ,σ,loc (Ω) → Hτ,σ,loc (Ω), preserving compactness of supports. Then for P there exists a properly supported parametrix Q = q(x, D), q(x, ξ) ∈ −m Sτ,σ,cl (Ω); namely QP = I + R1 , P Q = I + R2 , where R1 and R2 have symbols −∞ in Sτ,σ (Ω). m be elliptic in a neighborhood of x0 . Corollary 3.3. Let p(x, ξ) ∈ Sτ,σ,cl −∞ s s+m Then u ∈ Hτ,σ , P u ∈ Hτ,σ (x0 ) imply u ∈ Hτ,σ (x0 ).

The proof is by Theorem 3.3, Corollary 3.1 and Theorem 3.1. Using Corollary 3.2 and constructing microlocal parametrices, we deduce similarly the following micro-regularity result.

12


m Corollary 3.4. Let p(x, ξ) ∈ Sτ,σ,cl satisfy pm (x0 , ξ 0 ) 6= 0 for some −∞ s s+m x0 ∈ Ω, ξ 0 6= 0. Then u ∈ Hτ,σ , P u ∈ Hτ,σ (x0 , ξ 0 ) imply u ∈ Hτ,σ (x0 , ξ 0 ).

Comments 3.1. In the classes presented here we require only C ∞ regularity with respect to ξ. The corresponding symbols are comparable, for example, with those in Taniguchi [18] defined by the estimates (3.11)

|Dxα Dξβ p(x, ξ)| ≤ cβ M −|α| α!σ hξim−|β| .

m Namely, if (3.11) is satisfied we have p(x, ξ) ∈ Sτ,σ for a small τ > 0. We point out that, with respect to the standard calculus requiring also Gevrey estimates in ξ, cf. Rodino [16] and the references there, our present −∞ frame. More precisely, if regularizing operators R are such only in the Hτ,σ −∞ −∞ ∞ R = r(x, D) with r ∈ Sτ,σ , we have R : Hτ,σ → Hτ,σ , but for f ∈ E 0 (IRn ) ∞ n ∞ or even f ∈ C0 (IR ), in general Rf ∈ C is not of Gevrey class, neither the possible Gevrey local regularities and micro-regularities of f are preserved m under applications of R or P = p(x, D) with symbol p ∈ Sτ,σ .

4.

Gevrey paraproduct calculus n/2+ε

Let a ∈ H|τ |,σ , ε > 0, τ ∈ IR, σ > 1. We can define the paraproduct operator Ta as follows: X s (4.1) Ta u = (Sq a)uq , u ∈ Hτ,σ , q

P where {uq }∞ q=−1 denotes the dyadic decomposition of u, Sq a = −1≤p≤q−N1 ap , {ap } the dyadic decomposition of a. Let N1 be sufficiently large, cf. [4],[5], then we have s s Theorem 4.1. Ta : Hτ,σ → Hτ,σ is a continuous mapping for every s ∈ −∞ s s IR. Moreover, u ∈ Hτ,σ and u ∈ Hτ,σ (x0 ) imply Ta u ∈ Hτ,σ (x0 ) for any s ∈ IR n 0 s ,H s ) ≤ Cs kak n/2+ε . Fix further ξ and x0 ∈ IR . We have kTa kL(Hτ,σ 6= 0. If τ,σ H |τ |,σ

s t t u ∈ Hτ,σ (x0 ) with s > 0, then u ∈ Hτ,σ (x0 , ξ 0 ) implies Ta u ∈ Hτ,σ (x0 , ξ 0 ) for s < t < s + ε.

Proof. The same statement was already proved in [5, Theorem 3.1]. We think however it is worth to give in the following a precise argument for the −∞ s s pseudo-local property, i.e. u ∈ Hτ,σ and u ∈ Hτ,σ (x0 ) imply Ta u ∈ Hτ,σ (x0 ) for every s ∈ IR, since details in this connection are missing in [5]. Our present proof will be based on the pseudo-differential calculus of the preceding Section 3. 0 Let us assume for simplicity τ > 0. We may take φ ∈ Gσ0 (IRn ), 1 < σ 0 < σ, s with φ(x) = 1 in a neighborhood Vx0 of x0 , such that φu ∈ Hτ,σ . Let us then s σ0 show that φ1 Ta u ∈ Hτ,σ for every φ1 ∈ G0 (Vx0 ). In fact φ1 Ta u = φ1 Ta φu + φ1 Ta (1 − φ)u


13

s s where φ1 Ta φu ∈ Hτ,σ , granted the boundedness of Ta on Hτ,σ . h Let us prove that φ1 Ta (1 − φ)u ∈ Hτ,σ for all h ∈ IR. Basing on (4.1), we write X (4.2) φ1 Ta (1 − φ)u = φ1 (Sp a)ϕ(2−p D)(1 − φ)u + f, p≥0 +∞ where f ∈ Hτ,σ and ϕ is defined as in Lemma 2.2. Since φ1 ϕ(2−p ξ) and φ(x) 0 are symbols in Sτ,σ , we may apply Theorem 3.2 and write, with N to be fixed later: X φ1 ϕ(2−p D)(1−φ) = φ1 (α!)−1 Dxα (1−φ)2−p|α| (∂ξα ϕ)(2−p D)+φ1 rpN (x, D) |α| 0 and applying (d) in Theorem 2.1, we may limit ourselves to check that

(4.3)

kDα (Sp a)rpN (x, D)ukL2τ,σ ≤ cpα 2−ph+p|α| ,

{cpα }p ∈ l2 .

By Leibniz formula and Theorem 2.5, we are further reduced to prove the same estimates for the terms (4.4)

kDα1 (Sp a)kH n/2+ε0 kDα2 rpN (x, D)ukL2τ,σ , τ,σ

α1 + α2 = α,

with 0 < ε0 < ε. Now from the definition of Sp we have easily (4.5)

kDα1 (Sp a)kH n/2+ε0 ≤ c0p,α1 2p|α1 | kakH n/2+ε , τ,σ

τ,σ

{c0pα1 }p ∈ l2 .

It will be then convenient to write the explicit expression of rpN α2 (x, ξ), the symbol of Dα2 rpN (x, D). Namely, according to the last part of the proof of Theorem 3.2: X NZ 1 (4.6) rpN α2 (x, ξ) = rpN α2 γ (x, ξ, t)(1 − t)N −1 dt, γ! 0 |γ|=N

where rpN α2 γ is a linear combination of terms of the form Z −n ˆ (4.7) e(x, ξ, t) = (2π) eixζ 2−pN (∂ξγ ϕ)(2−p (ξ + tζ))ξ β1 ζ γ+β2 φ(ζ)dζ. with β1 + β2 = α2 .

14


We want to estimate the norm of the corresponding operator as a map h0 from Hτ,σ to L2τ,σ . Going back to Lemma 3.1 and to the proof of Theorem 3.1, we have then to consider ˆ eˆ(λ, ξ, t) = 2−pN (∂ξγ ϕ)(2−p (ξ + tλ))ξ β1 λγ+β2 φ(λ) and evaluate the L2 -norm with respect to λ of g(λ, ξ, t) = eτ λ

1/σ

0

hλiM eˆ(λ, ξ, t)hξi−h ,

where M is determined as in the proof of Theorem 3.1, depending on h0 . Assuming without loss of generality h0 < 0, we have 0

|(∂ξγ ϕ)(2−p (ξ + tλ))| ≤ cγ h2−p (ξ + tλ)ih −|α2 | 0

0

≤ cγ 2−ph +p|α2 | hξ + tλih −|α2 | 0

0

0

≤ c0γ 2−ph +p|α2 | hξih −|α2 | htλi−h +|α2 | and moreover for some δ > 0 ˆ |φ(λ)| ≤ ce−δλ

1/σ 0

,

so we obtain 0

sup sup kg(λ, ξ, t)kL2 (IRnλ ) ≤ c2−pN −ph +p|α2 |

0≤t≤1

ξ∈IRn

for a constant c independent of p. In view of (4.6), (4.7) and Lemma 3.1, we deduce that 0

kDα2 rpN (x, D)ukL2τ,σ ≤ c0α2 2−pN −ph +p|α2 | kukHτ,σ h0 and therefore from (4.4) and (4.5) 0

kDα ((Sp a)rpN (x, D)u)kL2τ,σ ≤ c00pα 2−pN −ph +p|α| kukHτ,σ h0 where {c00pα }p ∈ l2 . To obtain (4.3) it will be then sufficient to fix N > h − h0 . This concludes the proof of the pseudo-local property. For the other statements in Theorem 4.1 we refer to the proof in [5]. n/2+ε

(τ < 0), then in the the same way we can Remark 4.1. If a ∈ Hτ,σ s define the paraproduct operator Ta , which is a continuous mapping from H|τ |,σ s to Hτ,σ . −∞ s Remark 4.2. Observe in Theorem 4.1 that u ∈ Hτ,σ , u ∈ Hτ,σ (x0 ) s imply Ta u ∈ Hτ,σ (x0 ) without any restriction on s ∈ IR, whereas the microlocal statement depends on the local regularity of u. In fact when τ = 0 the paraproduct Ta belongs to the Hörmander’s class L01,1 , cf. [4], and it is well known that the corresponding pseudo-differential operators are pseudo-local but not micro-local in general.


15

From (4.1), the definition of Ta seems dependent on the dyadic decomposis tion of Gevrey-Sobolev space Hτ,σ (i.e. depending on the choice of {K, ϕ, N1 }). We suppose there exist two dyadic decompositions which depend on {K, ϕ, N1 } and {K 0 , ϕ0 , N10 } respectively, and denote by Ta and Ta0 as two paraproducts corresponding to {K, ϕ, N1 } and {K 0 , ϕ0 , N10 } respectively, then we have n/2+ε

s s+ε1 If a ∈ H|τ |,σ , then Ta − Ta0 ∈ L(Hτ,σ , Hτ,σ ), for any

Theorem 4.2. 0 < ε1 < ε, and

kTa − Ta0 kL(H s

(4.8)

s+ε1 τ,σ ,Hτ,σ )

≤ Cs kakH n/2+ε , |τ |,σ

P s Proof. Let P function aP(resp. u ∈ Hτ,σ )Phave two decompositions ap P P up and vp ), and Sq a = p≤q−N1 ap , Sq0 a = p≤q−N 0 a0p , and a0p (resp. 1 then (4.9) Ta u − Ta0 u =

X X q

X X q



X p

=

X p

ap 

X X q

p≤q−N1

+

=

ap uq −

ap vq −

X

(Sq0 a)vq

q

p≤q−N1

X

ap vq

p≤q−N1



(uq − vq ) +

ap ωp +



q

q≥p+N1

X



 X

X

ap − Sq0 a vq

p≤q−N1

ω˜q .

q

Without loss of generality, we let K 0 > K, then 0 supp ω ˆ p ⊂ Cp+N , 1

kωp kL2τ,σ ≤ cp 2−ps .

00 So if we choose N1 large enough, we have supp{ad p ωp } ⊂ Cp+N1 , and

kap ωp kL2τ,σ ≤ Ckap kH n/2+ε0 kωp kL2τ,σ |τ |,σ

p(ε0 −ε)

≤ C2

kap kH n/2+ε cp 2−ps , |τ |,σ

where ε0 ∈ (0, ε), then (4.10)

kap ωp kL2τ,σ ≤ c˜p kap kH n/2+ε 2−p(s+ε1 ) , |τ |,σ

where ε1 = ε − ε0 ∈ (0, ε), {˜ cp } ∈ l2 . c Next we have, for N1 large enough, that supp ω ˜q ⊂ Cq0 , and for any ε0 ∈ (0, ε), we have

16


(4.11) kω˜q kL2τ,σ

° ° ° ° X ° °  ° a − a ≤ ° p° ° ° ° p≤q−N1

  + ka − Sq0 akH n/2+ε0  kvq kL2τ,σ |τ |,σ

n/2+ε0

H|τ |,σ 0

≤ CkakH n/2+ε 2−q(ε−ε ) c0q 2−qs |τ |,σ

= c˜0q kakH n/2+ε 2−q(s+ε1 ) , ε1 = ε − ε0 ∈ (0, ε), {˜ c0q } ∈ l2 . |τ |,σ

s+ε1 This implies that Ta u − Ta0 u ∈ Hτ,σ , for any 0 < ε1 < ε, and the estimate (4.8) is obvious from the process above. Theorem 4.2 is proved. s s+ε1 From Theorem 4.2, we know Ta ≡ Ta0 (modL(Hτ,σ , Hτ,σ )). If we de−ε note Lτ,σ as the Gevrey ε-regular operator class, i.e. A ∈ L−ε τ,σ means A ∈ s s+ε 1 L(Hτ,σ , Hτ,σ ) for any s ∈ IR, then we also have Ta ≡ Ta0 (modL−ε τ,σ ), or −ε1 0 Ta − Ta ∈ Lτ,σ . We have the following composition result for the paraproduct operators: ε+n/2

Theorem 4.3. Let a, b ∈ Hτ,σ , τ ≥ 0, σ > 1, ε > 0, thus (see ε+n/2 Theorem 2.6 above) ab ∈ Hτ,σ . Then for any 0 < ε1 < ε, Ta ◦ Tb − Tab ∈ −ε1 s+ε1 ≤ Ckak n/2+ε kbk n/2+ε . Lτ,σ , and we have kTa ◦ Tb − Tab kL(H s ,Hτ,σ ) H H τ,σ

τ,σ

τ,σ

s Proof. Let u ∈ Hτ,σ , {ap }, {bp } and {up } the L-P decompostions with s respect to P a, b and u. Then we know from P Theorem 4.1 that v = Tb u ∈ Hτ,σ , 0 and v = vq , supp vˆqP ⊂ Cq ; Sq a = p2 ≤q−N1 ap2 . Then Ta ◦ Tb u = Ta v = P p1 ≤q−N1 bp1 uq , i.e. q (Sq a)vq + Rv, vq =

(4.12)

Ta ◦ Tb u =

X q

X

X

ap2 bp1 uq + R(Tb u).

p1 ≤q−N1 p2 ≤q−N1

q d Since supp S ˆq ⊂ Cq0 and Cq ⊂ Cq0 , then it is similar q a ⊂ B(0, C2 ), supp v to the proof of (4.9), we have easily

(4.13)

s+ε1 R(Tb u) ∈ Hτ,σ ,

0 < ε1 < ε.

Now we let (4.14)

X

dq =

X

ap2 bp1 .

p1 ≤q−N1 p2 ≤q−N1

Observe supp dˆq ⊂ B(0, C2q ), and ab − dq =

X p1 >q−N1 or p2 >q−N1

ap2 bp1 ,


17

we have, by Schauder-Gevrey estimate and assuming as before p1 > q − N1 or p2 > q − N1 in the sums, that X (4.15) k(ab − dq )uq kL2τ,σ ≤ kap2 bp1 kH n/2+ε0 kuq kL2τ,σ τ,σ X 0 ≤C kap2 kH n/2+ε kbp1 kH n/2+ε0 kuq kL2 0 , ε0 ∈ (0, ε) τ,σ τ,σ τ,σ ´ ³X −(p1 +p2 )ε1 2 cq 2−qs , ε1 = ε − ε0 ≤ CkakH ε+n/2 kbkH ε+n/2 τ,σ

τ,σ

≤ c˜q kakH n/2+ε kbkH n/2+ε 2−q(s+ε1 ) . τ,σ

τ,σ

s Thus it is similar to the proof of (4.9), we have, for u ∈ Hτ,σ , that X s+ε1 dq uq ∈ Hτ,σ Tab u − , q

P s+ε1 this means that Ta ◦ Tb u − Tab u = R(Tb u) + [ q dq uq − Tab u] ∈ Hτ,σ . The result on norm-estimate of composition may be easily checked from the proof process above. Theorem 4.3 is proved. With respect to the L2 -scalar product we can define the conjugation s s operator Ta∗ for paraproduct Ta : Hτ,σ → Hτ,σ by (4.16)

−s s hTa∗ u, vi = hu, Ta vi, u ∈ H−τ,σ , v ∈ Hτ,σ .

−s −s s s So Ta∗ : H−τ,σ → H−τ,σ = (Hτ,σ )0 (the dual space of Hτ,σ ). More precisely we have n/2+ε

Theorem 4.4. Let a ∈ H|τ |,σ (ε > 0, τ ∈ IR and σ > 1), then Ta∗ is 1 also a paraproduct operator and Ta∗ − Ta¯ ∈ L−ε τ,σ , for any ε1 ∈ (0, ε), and kTa∗ − Ta¯ kL(H s


−(s+ε1 )

s Proof. Let u ∈ Hτ,σ , v ∈ H−τ,σ

≤ CkakH n/2+ε . |τ |,σ

, then we have

h(Ta∗ − Ta¯ )u, vi = hTa∗ u, vi − hTa¯ u, vi =hu, Ta vi − hTa¯ u, vi, where hu, Ta vi =

X X Z

uq a ¯p v¯r dx,

q,r p≤r−N1

hTa¯ u, vi =

X X Z

a ¯p uq v¯r dx.

r,q p≤q−N1

P d 0 Observe supp( p≤r−N ap vT ˆq ⊂ Cq , and there exists N2 > 0, r ) ⊂ Cr , supp u 1 0 large enough, such that Cq Cr = ∅ if |q − r| > N2 . Hence if |q − r| > N2 , we have Z Z ˆq (−η)ad uq (x)(ap vr )dx = u p vr (η)dη = 0.

18


This implies hTa∗ u, vi =

X q

hTa¯ u, vi =

X q

X Z

X

uq a ¯p v¯r dx,

q−N2 ≤r≤q+N2 p≤r−N1

X Z

X

a ¯p uq v¯r dx.

q−N2 ≤r≤q+N2 p≤q−N1

Thus there exists a large integer N3 , such that X X |hTa∗ u, vi − hTa¯ u, vi| ≤ q

X

kap uq vr kL1 .

q−N2 ≤r≤q+N2 q−N3 ≤p≤q+N3

Let p = q +j1 , r = q +j2 , then by Cauchy-Schwarz inequality and Theorem 2.5, we have for ε0 = ε − ε1 ∈ (0, ε) |hTa∗ u, vi − hTa¯ u, vi| X X ≤ q

X

kaq+j1 kH n/2+ε0 kuq kL2τ,σ kvq+j2 kL2−τ,σ . |τ |,σ

|j2 |≤N2 |j3 |≤N3

Because N2 , N3 are finite and fixed, then we can further estimate by Ckaq kH n/2+ε 2−qε1 cq 2−qs c0q 2q(s+ε1 ) ≤ CkakH n/2+ε cq c0q , |τ |,σ

0

where ε1 = ε−ε ∈ (0, ε), and {cq }, CkvkH −(s+ε1 ) . Thus we obtain

|τ |,σ

{c0q }

2

0 s , k{c }kl2 ≤ ∈ l , k{cq }kl2 ≤ CkukHτ,σ q

−τ,σ

(4.17)

s kvk −(s+ε1 ) . |hTa∗ u, vi − hTa¯ u, vi| ≤ CkakH n/2+ε kukHτ,σ H |τ |,σ

−τ,σ

Theorem 4.4 is proved. From [5, Section 3], we also have the following paralinearization results: Theorem 4.5. Let F : C → C be an entire analytic function, and s s satisfy F (0) = 0. Let f be in Hτ,σ , s > n/2, τ > 0, σ > 1. Then F (f ) ∈ Hτ,σ t and F (f ) = TF 0 (f ) f + g, where g ∈ Hτ,σ for all t < 2s − n/2. Theorem 4.5 has the following obvious corollaries: Corollary 4.1. Let F : C → C be an entire analytic function, and let s f be in Hτ,σ (x0 ), s > n/2, τ > 0, σ > 1, x0 ∈ IRn . Then F (f ), which is s well defined in a neighborhood of x0 , belongs to Hτ,σ (x0 ). Fix further ξ 0 6= 0. t 0 t If f ∈ Hτ,σ (x0 , ξ ) for s < t < 2s − n/2, then also F (f ) ∈ Hτ,σ (x0 , ξ 0 ). P Corollary 4.2. Let F (x, z) = β cβ (x)z β , entire with respect to z, for 0 cβ ∈ Gσ (Ω) (1 < σ 0 < σ), z ∈ CN and x0 ∈ Ω ⊂ IRn , and let the components s s of f = (f1 , · · · , fN ) be in Hτ,σ (x0 ), s > n/2, τ > 0; then F (x, f ) ∈ Hτ,σ (x0 ). 0 σ After cutting off F and f by a function ϕ ∈ G0 (Ω) with ϕ(x) = 1 in a neighPN t borhood of x0 , we have F (x, f ) = j=1 T∂F/∂zj (x,f ) fj + g, where g ∈ Hτ,σ (x0 ) t for all t < 2s − n/2. If all the components of f are in Hτ,σ (x0 , ξ 0 ) for t s < t < 2s − n/2, ξ 0 6= 0, then also F (x, f ) ∈ Hτ,σ (x0 , ξ 0 ).


5.

19

Paradifferential operators in Gevrey classes In this section, m ∈ IR, σ > 1 as usual, but we shall assume τ > 0. Definition 5.1.

For ε > 0, let

m,ε lτ,σ ={l(x, ξ) | l is m order homogeneous C ∞ (IRn \ 0) function in ξ, n/2+ε and Hτ,σ function in x for ξ uniformly}. m,ε m,ε The functions l ∈ lτ,σ can be regarded as symbols in the classes Sτ,σ from Definition 3.1. Observe however that the corresponding pseudo-differential s operators l(x, D) are not Lm τ,σ class operators; in fact continuity from Hτ,σ s−m to Hτ,σ fails for large s, because of the limited Gevrey smoothness of l(x, ξ) with respect to x. Following Bony [4] and using the Gevrey paraproduct calculus of the preceding section, we shall then consider paradifferential operators associated to l(x, ξ), which will turn out to be of class Lm τ,σ . m,ε For l ∈ lτ,σ , we can define an operator Tl as X X s (Tl u)(x) = Sq (l(x, D))uq (x), u = uq ∈ Hτ,σ ,

Definition 5.2.

q

where Sq (l(x, D)) is the pseudo-differential operator with symbol Sq (l(x, ξ)), defined by letting Sq act on the x variables, cf. (4.1). P P If l(x, ξ) = j lj (x, ξ) is a finite sum, then we denote Tl = j Tlj . n/2+ε

If l(x, ξ) = a(x)h(ξ), a(x) ∈ Hτ,σ homogeneous, then (5.1)

, h(ξ) ∈ C ∞ (IRn \ 0) and m order

Sq (a(x)h(ξ)) = (Sq a)h(ξ).

Hence we have (Tl u)(x) =

X

Sq (a)(h(D)u)q ,

q

(h(D)u)q = (2π)−n

Z eixξ φ(2−q ξ)h(ξ)ˆ u(ξ)dξ = h(D)uq ,

i.e. (5.2)

(Tl u)(x) = Ta ◦ h(D)u,

if l = a(x)h(ξ).

m,ε For general l ∈ lτ,σ , we can rewrite

(5.3)

l(x, ξ) = |ξ|m l(x, ω), ω =

ξ ∈ S n−1 . |ξ|

˜ j (ω)} are correLet ∆ be Laplace-Beltrami operator on S n−1 , {λj } and {h ˜ j = λj h ˜ j ), we know {h ˜ j } is sponding eigenvalues and eigenfuctions (i.e. ∆h

20


a complete orthonormal basis in L2 (S n−1 ), and limj→∞ λj j −2/n ∈ (0, +∞). Since l(x, ω) ∈ L2 (S n−1 ), ω ∈ S n−1 , we have, by using Fourier expansion, that (5.4)

l(x, ω) =

X

˜ j (ω), aj (x)h

j

R

˜ j (ω)dω. where aj (x) = S n−1 l(x, ω)h Since ∆ is self-adjoint, we have Z ˜ j (ω)dω λkj aj (x) = l(x, ω)∆k h S n−1 Z ˜ j (ω)dω. ∆k l(x, ω)h = S n−1

Thus by Cauchy-Schwarz inequality, we have ¶ 12 µZ ˜ j (ω)kL2 (S n−1 ) |λj |k kaj (x)kH n/2+ε ≤ k∆k l(x, ω)k2H n/2+ε dω kh τ,σ

µZ

τ,σ

S n−1

=

k∆ S n−1

n/2+ε

Since ∆k l(x, ω) ∈ Hτ,σ

k

l(x, ω)k2H n/2+ε dω τ,σ

¶ 12 .

in x, hence we have obtained |λj |k kaj (x)kH n/2+ε τ,σ

n/2+ε

≤ Ck , {Ck } is a bounded constant set. This implies aj (x) ∈ Hτ,σ 2

kaj kH n/2+ε ≤ Ck j − n k ,

(5.5)

τ,σ

, and

∀k,

is rapidly decreasing in j. On the other hand from Sobolev lemma, we have for an even integer s1 , satisfying s1 > n/2 + M ˜ j (ω)kC M (S n−1 ) ≤ Ckh ˜ j (ω)kH s1 (S n−1 ) kh s1 /2

≤C

X

˜ j (ω)kL2 (S n−1 ) k∆k h

k=0 s1 /2

≤C

X

|λj |k .

k=0

That is (5.6)

˜ j (ω)kC M (S n−1 ) ≤ CM j kh

(M +n/2+1) n

,

is temperedly increasing in j. Actually we have proved the following result: Lemma 5.1. decomposition (5.7)

m,ε Let l ∈ lτ,σ , then l has the following spherical harmonic

l(x, ξ) =

X j

aj (x)hj (ξ),


21

n/2+ε

where aj (x) ∈ Hτ,σ and kaj kH n/2+ε is rapidly decreasing in j, hj (ξ) = τ,σ ˜ j (ξ/|ξ|) and kh ˜ j (ω)kC M (S n−1 ) is temperedly increasing in j for any M |ξ|m h fixed. Since (hj (D)u)q = hj (D)uq , now we can define the operator Tl as follows: X s (5.8) Tl u = Taj ◦ hj (D)u, u ∈ Hτ,σ , j s ,H s ) ≤ Ckaj k n/2+ε is rapidly decreasing in j, and the norm where kTaj kL(Hτ,σ τ,σ Hτ,σ of hj (D)u is temperedly increasing, then the series (5.8) is convergent. We can prove Tl , as defined by (5.8), is Lm τ,σ class operator, i.e.

Theorem 5.1. linear operator.

m,ε s s−m For l ∈ lτ,σ , Tl : Hτ,σ → Hτ,σ (∀s ∈ IR) is a bounded

Proof. We may write XX (5.9) Tl u = Sq (aj )hj (D)uq , j

s u ∈ Hτ,σ ,

q

d q ⊂ Cq , supp Sqd where supp hj (D)u (aj ) ⊂ B(0, K2q−N1 ). So for N1 large enough we have q−N1 d d d supp Sq (aj )h ) ⊂ Cq0 , j (D)uq = supp Sq (aj ) ∗ hj (D)uq ⊂ Cq + B(0, K2

dD))uq ⊂ C 0 . Thus i.e. supp Sq (l(x, q

Z Sq (l(x, D))uq (x) = (2π)−n eixξ Sq (l(x, ξ))ˆ uq (ξ)dξ X = Sq (aj )hj (D)uq , j

and kSq (l(x, D))uq (x)kL2τ,σ ≤

X

kSq (aj )hj (D)uq kL2τ,σ

j

≤

X

kSq (aj )kH n/2+ε khj (D)uq kL2τ,σ τ,σ

j

≤

X

kaj kH n/2+ε khj (D)uq kL2τ,σ , τ,σ

j

where 1

khj (D)uq kL2τ,σ = k exp(τ hξi σ )hj (ξ)ˆ uq kL2 1 ˜ j (ξ/|ξ|)|ξ|m u = k exp(τ hξi σ )h ˆq kL2 (q+1) m ˜ ≤ khj (ω)kC(S n−1 ) (K2 ) kˆ uq kL2

τ,σ

˜ j (ω)kC(S n−1 ) cq 2−q(s−m) . ≤ kh

22


˜ j (ω)kC(S n−1 ) is temperedly Since kaj kH n/2+ε is rapidly decreasing in j and kh τ,σ increasing in j, we have kSq (l(x, D))uq kL2τ,σ ≤ Ccq 2−q(s−m) ,

(5.10)

{cq } ∈ l2 .

P s , then we have proved Since Tl u = q Sq (l(x, D))uq , and k{cq }kl2 ≤ C1 kukHτ,σ s−m s−m ≤ CC1 . Theorem 5.1 is proved. Tl u ∈ Hτ,σ , and kTl kL(H s ,Hτ,σ ) τ,σ

It seems, from Definition 5.2, the operator Tl depends on the dyadic decomposition. However if {K 0 , φ1 , N10 } is another dyadic decomposition, and Tl0 is the corresponding operator, then Tl − Tl0 =

X

(Taj − Ta0 j ) ◦ hj (D).

j

We have proved in Theorem 4.2 that 0 1 Taj − Ta0 j ∈ L−ε τ,σ , ε1 ∈ (0, ε), and kTaj − Taj kL(H s


≤ Ckaj kH n/2+ε , τ,σ

s s−m and hj (D) : Hτ,σ → Hτ,σ . Thus it is easy to prove that 1 Tl − Tl0 ∈ Lm−ε τ,σ , ∀ε1 ∈ (0, ε).

(5.11)

Let us consider the composition of two operators. Theorem 5.2. l(x, ξ) =

mk ,ε Let lk (x, ξ) ∈ lτ,σ (k = 1, 2), ε ∈ \ IN, and

X |α| 0 and ε ∈ \ IN. Then

n/2+ε Hτ,σ ,

R = h(D) ◦ Ta −

X |α| n + ε/2, we obtain easily ˜ C 2M (S n−1 ) . k(1 + |x|ε )r(x)kL1 (IRn ) ≤ Ckhk

(5.12)

s For u ∈ Hτ,σ , we have  X X Ru = 2mq h1 (2−q D)Sq (a) − q

|α| 0, we know R(a, v) ∈ Hτ,σ , for any ε1 ∈ (m, ε). P P 0 ˆ Also Tv a = q Sq (v)aq = fq , where supp fq ⊂ supp a ˆq + supp(Sd q (v)) ⊂ Cq , and kfq kL2τ,σ ≤ kaq kH n/2+ε0 kSq (v)kL2τ,σ (for ∀ε0 ∈ (0, ε)) τ,σ X ≤ kaq kH n/2+ε0 kvp kL2τ,σ τ,σ

p≤q−N1 q(n/2+ε0 )

≤ kaq kL2τ,σ 2 ≤

0 c0q 2−q(ε−ε )

X

X

cp 2−p(s−m)

p≤q−N1

cp 2−p(s−m) .

p≤q−N1

P ε−ε0 for any If s > m, then p cp 2−p(s−m) ≤ C < ∞, which implies Tv a ∈ Hτ,σ 0 0 0 0 −q(ε−ε ) −q(s−m) 00 −q(s+ε−ε −m) ε ∈ (0, ε). If s < m, then kfq kL2τ,σ ≤ cq 2 C2 = cq 2 , 0

s+ε−ε −m i.e. Tv a ∈ Hτ,σ for any ε0 ∈ (0, ε). If s = m, then we have kfq kL2τ,σ ≤ 0

0

ε−ε Cc0q 2−q(ε−ε ) kvkL2τ,σ , i.e. Tv a ∈ Hτ,σ for any ε0 ∈ (0, ε). Therefore we have 0

s proved Tv a ∈ Hτ,σ for s0 < min{ε, s + ε − m}. Theorem 5.4 is proved.

Since

T

m,ε ε lτ,σ

m ⊂ Sτ,σ , from Theorem 5.4, Corollaries 3.1 and 3.2 we get:

m,ε Corollary 5.1. If l ∈ lτ,σ for all ε > 0, then l(x, D) − Tl is s s0 “regularizing” operator, i.e. l(x, D) − Tl ∈ L(Hτ,σ , Hτ,σ ) for any s and s0 . s0 s0 −m s0 Thus u ∈ Hτ,σ (x0 ) implies Tl u ∈ Hτ,σ (x0 ), and u ∈ Hτ,σ (x0 , ξ 0 ) implies 0 s −m Tl u ∈ Hτ,σ (x0 , ξ 0 ).

Applying further Corollary 3.4, we deduce m,ε s Corollary 5.2. Let l ∈ lτ,σ , ε > m, ε ∈ \ IN. If u ∈ Hτ,σ , then u ∈ t 0 0 implies Tl u ∈ Hτ,σ (x0 , ξ ) for t = min{s + [ε] − m, s − m}.

s0 Hτ,σ (x0 , ξ 0 )

Theorem 5.5. have (5.14)

m,ε Let l ∈ lτ,σ , ε > 0, then for the symbol σ(Tl ) of Tl , we

k∂ξα ∂xβ σ(Tl )kL2τ,σ ≤ Cαβ (1 + |ξ|)m−|α|+|β|−n/2−ε , m−n/2−ε

that is σ(Tl ) ∈ Sτ,σ,1,1

, cf. Definition 3.4.


27 n/2+ε

Proof. Without loss of generality, we take l(x, ξ) = a(x)h(ξ), a ∈ Hτ,σ and order of h is m. Then X σ(Tl )(x, ξ) = Sq (l(x, ξ))ϕ(2−q ξ) q

=

X

Sq (a)(x)h(ξ)ϕ(2−q ξ).

q

Thus ∂ξα ∂xβ σ(Tl )(x, ξ) = k∂ξα ∂xβ σ(Tl )(·, ξ)kL2τ,σ

≤

X

X

∂xβ Sq (a)(x)∂ξα (h(ξ)ϕ(2−q ξ)),

q

k∂xβ Sq (a)kL2τ,σ |∂ξα h(ξ)ϕ(2−q ξ)|

q

≤

X X q

k∂xβ ap kL2τ,σ Cα

X

µ |h(α1 )

α1 +α2 =α

p≤q−N1

ξ |ξ|

¶ ||ξ|m−|α1 | 2−q|α2 | |ϕ(α2 ) (2−q ξ)|.

Then, from Theorem 2.1, k∂ξα ∂xβ σ(Tl )(·, ξ)kL2τ,σ X ˜ C α (S n−1 ) 2q(m−|α|) |ϕ(α2 ) (2−q ξ)| ≤ C˜α cqβ 2q[|β|−(n/2+ε)] khk q

≤

C˜α0

X

cqβ 2q(m−|α|+|β|−n/2−ε) |ϕ(2−q ξ)|,

q

where {cqβ }q ∈ l2 . Since on supp ϕ(2−q ξ), |ξ| ≈ 2q , then the estimate above implies that (5.14) holds. Next, let Ω ⊂ IRn be an open subset, m ∈ IR, ε > 0, and ε ∈\ IN, we define Gevrey paradifferential symbol class Σm,ε τ,σ (Ω) as follows: Definition 5.3. (5.15)

We call Γ(x, ξ) ∈ Σm,ε τ,σ (Ω), if

Γ(x, ξ) = Γm (x, ξ) + Γm−1 (x, ξ) + ... + Γm−[ε] (x, ξ),

where Γm−k (x, ξ) is C ∞ (IRn \ 0) and m − k order homogeneous in ξ, and is n/2+ε−k Hτ,σ,loc (Ω) in x for ξ uniformly. 1 2 m1 +m2 ,ε k ,ε If Γk ∈ Σm (Ω) as τ,σ (Ω), (k = 1, 2), we define Γ #Γ ∈ Στ,σ

(5.16)

Γ1 #Γ2 =

X |α|+k1 +k2