Pseudo-differential calculus in anisotropic Gelfand-Shilov setting

We study some classes of pseudo-differential operators with symbols $a$ admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type S. Moreover, we deduce algebraic and certain invariance properties of these classes.


Introduction
Gelfand-Shilov spaces of type S have been introduced in the book [16] as an alternative functional setting to the Schwartz space S (R d ) of smooth and rapidly decreasing functions for Fourier analysis and for the study of partial differential equations. Namely, fixed s > 0, σ > 0, the space S σ s (R d ) can be defined as the space of all functions f ∈ C ∞ (R d ) satisfying an estimate of the form for some constant h > 0, or the equivalent condition for some constants h, r > 0. For σ > 1, S σ s (R d ) represents a natural global counterpart of the Gevrey class G σ (R d ) but, in addition, the condition (0.2) encodes a precise description of the behavior at infinity of f . Together with S σ s (R d ) one can also consider the space Σ σ s (R d ), which has been defined in [25] by requiring (0.1) (respectively (0.2)) to hold for every ε > 0 (respectively for every h, r > 0). The duals of S σ s (R d ) and Σ σ s (R d ) and further generalizations of these spaces have been then introduced in the spirit of Komatsu theory of ultradistributions, see [14,25].
After their appearance, Gelfand-Shilov spaces have been recognized as a natural functional setting for pseudo-differential and Fourier integral operators, due to their nice behavior under Fourier transformation, and applied in the study of several classes of partial differential equations, see e. g. [1,[3][4][5][6][7][8].
According to the condition on the decay at infinity of the elements of S σ s (R d ) and Σ σ s (R d ), we can define on these spaces pseudo-differential operators with symbols a(x, ξ) admitting an exponential growth at infinity. These operators are commonly known as operators of infinite order and they have been studied in [2] in the analytic class and in [12,24,34] in the Gevrey spaces where the symbol has an exponential growth only with respect to ξ and applied to the Cauchy problem for hyperbolic and Schrödinger equations in Gevrey classes, see [12,13,15,23]. Parallel results have been obtained in Gelfand-Shilov spaces for symbols admitting exponential growth both in x and ξ, see [3,4,7,8,11,27]. We stress that the above results concern the non-quasi-analytic isotropic case s = σ > 1. In [10] we considered the more general case s = σ > 0, which is interesting in particular in connection with Shubintype pseudo-differential operators, cf. [5,9]. Although the extension of the complete calculus developed in [3,4] in this case is out of reach due to the lack of compactly supported functions in S σ s (R d ) and Σ σ s (R d ), nevertheless some interesting results can be achieved also in this case by using different tools than the usual micro-local techniques, namely a method based on the use of modulation spaces and of the short time Fourier transform.
In the present paper, we further generalize the results of [10] to the case when s > 0 and σ > 0 may be different from each other. Thus the symbols we consider may have different rates of exponential growth and anisotropic Gevrey-type regularity in x and ξ. More precisely, the symbols should obey conditions of the form for suitable restrictions on the constants h, r > 0 (cf. (0.2)). We prove that if h > 0, and (0.3) holds true for every r > 0, then the pseudodifferential operator Op(a) is continuous on S σ s and on (S σ s ) ′ . If instead r > 0, and (0.3) holds true for every h > 0, then we prove that Op(a) is continuous on Σ σ s and on (Σ σ s ) ′ (cf. Theorems 3.7 and 3.13). We also prove that pseudo-differential operators with symbols satisfying such conditions form algebras (cf. Theorems 3.16 and 3.17). Finally we show that our span of pseudo-differential operators is invariant under the choice of representation (cf. Theorem 3. 6).
An important ingredient in the analysis which is used to reach these properties concerns characterizations of symbols above in terms of suitable estimates of their short-time Fourier transforms. Such characterizations are deduced in Section 2.
The paper is organized as follows. In Section 1, after recalling some basic properties of the spaces S σ s (R d ) and Σ σ s (R d ), we introduce several general symbol classes. In Section 2 we characterize these symbols in terms of the behavior of their short time Fourier transform. In Section 3 we deduce continuity on S s (R d ) and Σ s (R d ), composition and invariance properties for pseudo-differential operators in our classes.

Preliminaries
In this section we recall some basic facts, especially concerning Gelfand-Shilov spaces, the short-time Fourier transform and pseudo-differential operators.
We let S (R d ) be the Schwartz space of rapidly decreasing functions on R d together with their derivatives, and by S ′ (R d ) the corresponding dual space of tempered distributions. Moreover M(d, R) will denote the vector space of real d × d matrices.
s for some h > 0, ε > 0 (respectively for every h > 0, ε > 0). Moreover we recall that for s < 1 the elements of S σ s (R d ) admit entire extensions to C d satisfying suitable exponential bounds, cf. [16] for details.
The Gelfand-Shilov distribution spaces (S σ s ) ′ (R d ) and (Σ σ s ) ′ (R d ) are the projective and inductive limit respectively of (S σ s;h ) ′ (R d ). This means that We remark that in [26] it is proved that For every s, σ > 0 we have for every ε > 0. If s + σ ≥ 1, then the last two inclusions in (1.3) are dense, and if in addition (s, σ) = ( 1 2 , 1 2 ), then the first inclusion in (1.3) is dense.
From these properties it follows that . The Gelfand-Shilov spaces possess several convenient mapping properties. For example they are invariant under translations, dilations, and to some extent tensor products and (partial) Fourier transformations.
The Fourier transform F is the linear and continuous map on S (R d ), given by the formula Here · , · denotes the usual scalar product on R d . The Fourier transform extends uniquely to homeomorphisms from . Some considerations later on involve a broader family of Gelfand-Shilov spaces. More precisely, for s j , σ j ∈ R + , j = 1, 2, the Gelfand-Shilov spaces S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) and Σ σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) consist of all functions F ∈ C ∞ (R d 1 +d 2 ) such that for some h > 0 respective for every h > 0. The topologies, and the duals , respectively, and their topologies are defined in analogous ways as for the spaces S σ s (R d ) and Σ σ s (R d ) above. The following proposition explains mapping properties of partial Fourier transforms on Gelfand-Shilov spaces, and follows by similar arguments as in analogous situations in [16]. The proof is therefore omitted. Here, F 1 F and F 2 F are the partial Fourier transforms of F (x 1 , x 2 ) with respect to x 1 ∈ R d 1 and x 2 ∈ R d 2 , respectively. Proposition 1.1. Let s j , σ j > 0, j = 1, 2. Then the following is true: (1) the mappings F 1 and F 2 on S (R d 1 +d 2 ) restrict to homeomorphisms (2) the mappings F 1 and F 2 on S (R d 1 +d 2 ) are uniquely extendable to homeomorphisms . The same holds true if the S σ 1 ,σ 2 s 1 ,s 2 -spaces and their duals are replaced by corresponding Σ σ 1 ,σ 2 s 1 ,s 2 -spaces and their duals. The next two results follow from [14]. The proofs are therefore omitted.
Let φ ∈ S σ s (R d ) \ 0 be fixed. Then the short-time Fourier transform In the case f ∈ L p (R d ), for some p ∈ [1, ∞], then V φ f is given by The following characterizations of the S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ), Σ σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) and their duals follow by similar arguments as in the proofs of Propositions 2.1 and 2.2 in [31]. The details are left for the reader. Proposition 1.3. Let s j , σ j > 0 be such that s j + σ j ≥ 1, j = 1, 2, s 0 ≤ s and σ 0 ≤ σ. Also let φ ∈ S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) \ 0 and let f be a Gelfand-Shilov distribution on R d 1 +d 2 . Then the following is true: ( holds for some r > 0; holds for every r > 0. A proof of Proposition 1.3 can be found in e. g. [20] (cf. [20,Theorem 2.7]). The corresponding result for Gelfand-Shilov distributions is the following improvement of [30, Theorem 2.5].
Proposition 1.4. Let s j , σ j > 0 be such that s j + σ j ≥ 1, j = 1, 2, s 0 ≤ s and t 0 ≤ t. Also let φ ∈ S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) \ 0 and let f be a Gelfand-Shilov distribution on R d 1 +d 2 . Then the following is true: ( holds for every r > 0; holds for some r > 0. A function ω on R d is called a weight or weight function, if ω, 1/ω ∈ L ∞ loc (R d ) are positive everywhere. It is often assumed that ω is vmoderate for some positive function v on R d . This means that (1.9) If v is even and satisfies (1.9) with ω = v, then v is called submultiplicative. For any s > 0, let P s (R d ) (P 0 s (R d )) be the set of all weights ω on R d such that e −r|x| 1 s ω(x) e r|x| 1 s for some r > 0 (for every r > 0). In similar ways, if s, σ > 0, then P s,σ (R 2d ) (P 0 s,σ (R 2d )) consists of all submultiplicative weight functions ω on R 2d such that for some r > 0 (for every r > 0). In particular, if ω ∈ P s,σ (R 2d ) (P 0 s,σ (R 2d )), then for some r > 0 (for every r > 0).
be fixed, and let a ∈ S s (R 2d ). Then the pseudo-differential operator Op A (a) with symbol a is the continuous operator on S s (R d ), defined by the formula We set Op t (a) = Op A (a) when t ∈ R, A = t · I and I is the identity matrix, and notice that this definition agrees with the Shubin type pseudo-differential operators (cf. e. g. [29]).
(1.12) It is easily seen that the latter definition agrees with (1.11) ′ when a ∈ L 1 (R 2d ).
If t = 1 2 , then Op t (a) is equal to the Weyl operator Op w (a) for a. If instead t = 0, then the standard (Kohn-Nirenberg) representation a(x, D) is obtained.
1.4. Symbol classes. Next we introduce function spaces related to symbol classes of the pseudo-differential operators. These functions should obey various conditions of the form for functions on the phase space R 2d . For this reason we consider seminorms of the form indexed by h > 0, Definition 1.5. Let s, σ and h be positive constants, let ω be a weight on R 2d , and let is finite for every r > 0, and the topology is the projective limit topology of Γ σ,s;h (ωr) (R 2d ) with respect to r > 0; (2) The sets Γ σ,s (ω) (R 2d ) and Γ σ,s;0 (ω) (R 2d ) are given by and their topologies are the inductive respective the projective topologies of Γ σ,s;h (ω) (R 2d ) with respect to h > 0. Furthermore we have the following classes.
where the supremum is taken over all In order to define suitable topologies of the spaces in Definition 1.6, is a Banach space, and the sets in Definition 1.6 are given by and and we equip these spaces by suitable mixed inductive and projective limit topologies of (Γ σ 1 ,σ 2 s 1 ,s 2 ) (h,r) (R d 1 +d 2 ). In Appendix A we show some further continuity results of the symbol classes in Definition 1.6.

The short-time Fourier transform and regularity
In this section we deduce equivalences between conditions on the short-time Fourier transforms of functions or distributions and estimates on derivatives.
In what follows we let κ be defined as In the sequel we shall frequently use the well known inequality Then the following is true: ( satisfies for every h > 0 (resp. for some h > 0), then for every h > 0 (resp. for some new h > 0); for every h > 0 (resp. for some h > 0), then f ∈ C ∞ (R d ) and satisfies for every h > 0 (resp. for some new h > 0).
Proof. We only prove the assertion when (2.2) or (2.4) are true for every h > 0, leaving the straight-forward modifications of the other cases to the reader. Assume that (2.2) holds. Then for every x ∈ R d the function for every h, r 0 > 0. In particular, 3) follows from the second inequality in (2.5), and (1) follows. Next we prove (2). By the inversion formula we get Here we notice that is an integrable function for every x, α and β, giving that f in (2.6) is smooth. By differentiation and the fact that φ ∈ Σ σ s we get for every h 1 > 0 and h 2 > 0. Since we get and h 1 can be chosen arbitrarily large, it follows from the last estimate that for every h 2 > 0. This gives the result.
By similar arguments we get the following result. The details are left for the reader.
Then the following is true: ( for every h > 0 (resp. for some h > 0), then f ∈ C ∞ (R d 1 +d 2 ) and satisfies for every h > 0 (resp. for some new h > 0).
As a consequence of the previous result we get the following.
\ 0 and let f be a Gelfand-Shilov distribution on R d 1 +d 2 . Then the following is true: . By similar arguments that led to Proposition 2.2 we also get the following. The details are left for the reader.
We also have the following version of Proposition 2.1 ′ , involving certain types of moderate weights.
and let a be a Gelfand-Shilov distribution on R 2d . Then the following is true: (1) If a ∈ C ∞ (R 2d ) and satisfies

Proof.
We shall use similar arguments as in the proof of Proposition then the fact that ω(X) ω(Y + X)e r 0 (|y| 1 is a bounded set of Σ σ,s s,σ . Hence for every h, r > 0. In particular, for every r > 0. Since it follows that (2.10) holds for all r > 0. This gives (1) in the case when ω ∈ P s,σ (R 2d ) and φ ∈ Σ σ,s s,σ (R 2d ) \ 0. In the same way, (1) follows in the case when ω ∈ P 0 s,σ (R 2d ) and φ ∈ S σ,s s,σ (R 2d ) \ 0. The details are left for the reader.
Next we prove (2) in the case · · · . Therefore, suppose (2.10) holds for all r > 0. Then a is smooth in view of Proposition 2.1 ′ .
By differentiation, (2.6), the fact that ω(Z) ω(X)e r 0 (|x−z| 1 s +|ξ−ζ| 1 σ ) , and the fact that φ ∈ Σ σ,s s,σ we get for every h, r > 0. Here the last inequality follows from (2.7). It follows that (2.9) holds for every h > 0 by using the estimates above and similar computations as in (2.8). The remaining case follows by similar arguments and is left for the reader.
3. Invariance, continuity and composition properties for pseudo-differential operators In this section we deduce invariance, continuity and composition properties for pseudo-differential operators with symbols in the classes considered in the previous sections. In the first part we show that for any such class S, the set Op A (S) of pseudo-differential operators is independent of the matrix A. Thereafter we deduce that such operators are continuous on Gelfand-Shilov spaces and their duals. In the last part we deduce that these operator classes are closed under compositions.
3.1. Invariance properties. An important ingredient in these considerations concerns mapping properties for the operator e i AD ξ ,Dx . In fact we have the following.
By straightforward application of Leibniz rule in combination with (1.10) we obtain for every h > 0 and R > 0. This is the same as (2). It remains to prove that (3) implies (1), but this follows by similar arguments as in the proof of Proposition 2.1. The details are left for the reader.
By (3.5) we have where H = K 1 * G and K j = ω −1 jcR · F 0,a , j ≥ 1. By Minkowski's inequality, letting Y 1 = (y 1 , η 1 ) as variables of integration, we get By combining these estimates we get and the result follows.
Also let By straight-forward applications of Parseval's formula, we get (cf. Proposition 1.7 in [29] and its proof). This gives , and the result follows in this case. Here the third equivalence follows from the fact that ω 0,R+c ω t,R ω 0,R−c , for some c > 0.
We note that if A, B ∈ M(d, R) and a, b ∈ (S σ,s s,σ ) ′ (R 2d ) or a, b ∈ (Σ σ,s s,σ ) ′ (R 2d ), then the first part of the previous proof shows that The following result follows from Theorems 3.1 and 3.3. The details are left for the reader.

3.2.
Continuity for pseudo-differential operators with symbols of infinite order on Gelfand-Shilov spaces of functions and distributions. Next we deduce continuity for pseudo-differential operators with symbols in the classes in Definitions 1.5 and 1.6. We begin with the case when the symbols belong to Γ σ,s s,σ;0 (R 2d ).
For the proof we need the following result. Then and is a bounded sets in S σ s;h 3 (R d ). Proof.
Since Ω 1 is a bounded set in S σ s;h 1 (R d ), there are constants C > 0 such that for every f ∈ Ω 1 . We shall prove that (3.7) is true for all f ∈ Ω 2 for a new choice of C > 0, and h 2 in place of h 1 . Let f ∈ Ω 2 . Then for some constant C which is independent of f , and the assertion on Ω 2 follows.
The same type of arguments shows that is a bounded set in S σ s;h 2 (R d ), and the boundedness of Ω 3 in S σ s;h 3 (R d ) follows by combining the boundedness of Ω 2 and the boundedness of (3.8) in S σ s;h 2 (R d ). Lemma 3.9. Let s, τ > 0, and set

9)
for every ε > 0, and for some positive constant r which depends on d, s and τ only.

Proof.
We have for some r 0 > 0 depending only on d, s and τ . Let Then g k (t) ≤ g 0,k (t 1 s ), and the result follows if we prove C k h k 0 k! s . By straight-forward computations, it follows that the maximum of t sk e −r 0 t is attained at t = sk/r 0 , giving that where the last inequality follows from Stirling's formula. This gives the result.
Proof of Theorem 3.7. By Theorem 3.1 it suffices to consider the case A = 0, that is for the operator Observe that Let now h 1 > 0 and f ∈ Ω, where Ω is a bounded subset of S σ s,h 1 (R d ). For fixed α, β ∈ N d we get By Lemma 3.8 and the fact that (2j)! ≤ 2 j j! 2 , it follows that for some h > 0, for every r > 0. This implies that for some positive constants h and r 0 we get for every r > 0. Hence, for every r > 0, provided τ is chosen such that τ h < 1. By inserting this into (3.11) and using Lemma 3.9 we get for some h > 0 and some r 0 > 0 that provided that r above is chosen to be smaller than r 0 . Then the continuity of Op A (a) on S σ s (R d ) follows. The continuity of Op A (a) on (S σ s ) ′ (R d ) now follows from the preceding continuity and duality.
Next we shall discuss corresponding continuity in the Beurling case. The main idea is to deduce such properties by suitable estimates on short-time Fourier transforms of involved functions and distributions. First we have the following relation between the short-time Fourier transforms of the symbols and kernels of a pseudo-differential operator. and x − y) be the kernels of Op A (a) and Op A (φ), respectively. Then (3.12) The essential parts of (3.12) is presented in the proof of [32, Proposition 2.5]. In order to be self-contained we here present a short proof.
By formal computations, using Fourier's inversion formula we get where all integrals should be interpreted as suitable Fourier transforms. This gives the result.
By similar arguments we also get the following. The details are left for the reader.
where h a,x = K 0,a (x, · ), and let φ j ∈ Σ σ s (R d ) be such that φ j L 2 = 1, j = 1, 2. By Moyal's identity we get and applying the short-time Fourier transform on g and using Fubbini's theorem on distributions we get Now suppose that r > 0 is arbitrarily chosen. By Proposition 2.2 we get for some c ∈ (0, 1) which depends on s and σ only, that for some where r 2 only depends on r and r 0 . Since for some h > 0 depdending on r 2 , and thereby by r and r 0 only. This implies  Theorem 3.14. Let A ∈ M(d, R), s, σ > 0 be such that s + σ ≥ 1 and (s, σ) = ( 1 2 , 1 2 ), and let a ∈ Γ σ,s s,σ (R 2d ).

3.3.
Compositions of pseudo-differential operators. Next we deduce algebraic properties of pseudo-differential operators considered in Theorems 3.7, 3.13 and 3.14. We recall that for pseudo-differential operators with symbols in e. g. Hörmander classes, we have .
More generally, if A ∈ M(d, R) and a 1 # A a 2 is defined by for a 1 and a 2 belonging to certain Hörmander symbol classes, then it follows from the analysis in [21] that for suitable a 1 and a 2 . We recall that the map a → K a,A is a homeomorphism from S σ,s s,σ (R 2d ) to S σ s (R 2d ) and from Σ σ,s s,σ (R 2d ) to Σ σ s (R 2d ). It is also obvious that the map Here we have identified operators with their kernels. Since when K j ∈ L 2 (R 2d ), j = 1, 2, 3, and that , the following result follows from these continuity results and (3.15).

Appendix A
In what follows we prove some auxiliary results on continuity of Gevrey symbol classes.
The next result concerns mapping properties of Γ σ,s s,σ spaces under trace operators.