Pseudo-Differential Calculus in Anisotropic Gelfand–Shilov Setting

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


Introduction
In the paper we deduce algebraic and continuity properties for a family of anisotropic pseudo-differential operators of infinite orders when acting on Gelfand-Shilov spaces. We permit superexponential growth on corresponding symbols and deduce continuity properties in the full range of (classical) Gelfand-Shilov spaces. We also deduce that the operator classes are closed under compositions.
Pseudo-differential operators (as well as Fourier integral operators) with ultra-differentiable symbols a(x, ξ) which are permitted to grow faster than polynomials at infinity, are commonly known as operators of infinite order. Operators of infinite order appear naturally when dealing with various kinds of partial differential equations, usually emerging in science and engineering. Such operators have been studied in different ways, e. g. in [2][3][4][5][6][7][8][9]11,12,[15][16][17]29,35]. Keyparts of these investigations consist of deducing fundamental algebraic and continuity properties.
The assumptions on the symbols for pseudo-differential operators with infinite order are more extreme compared to classical pseudo-differential operators. For the symbols to operators of infinite order, stronger regularity are imposed while growth conditions are relaxed compared to symbols of classical operators (Gevrey regularity and exponential type bound conditions instead of smoothness and polynomial bound conditions).
In order to meet the more extreme conditions on symbols to operators of infinite order, the spaces of Schwartz functions and their distribution spaces, feasible when dealing with classical pseudo-differential operators, are replaced by Gelfand-Shilov spaces and their distribution spaces. For fixed s, σ > 0, the Gelfand- for some (for every) h, r > 0. (See [22] and Sect. 1 for notations.) 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 (0.1) encodes a precise description of the behavior at infinity of f .
Continuity properties for operators of infinite order are important when investigating well-posedness for partial differential equations in the framework of Gelfand-Shilov spaces. Some studies are performed in [2,11,25,35] where the symbols have exponential growth with respect to the momentum variable. In [11,12,14,24,25] such operators are applied to Cauchy problems for hyperbolic and Schrödinger equations in Gevrey classes. Parallel results have also been obtained in Gelfand-Shilov spaces (see [3,4,7,8,10,29]). In the latter case, the symbols of the involved operators of infinite order admit exponential growth both in configuration and momentum variables, i. e. in the phase space variables.
For pseudo-differential operators of infinite order, their symbols should obey conditions of the form (0. 2) or, what seems to be more general, for some positive constants s, s j , σ, σ j and some positive function ω(x, ξ) defined on the phase space R 2d , j = 1, 2. A common condition is that ω should be moderate, meaning that it exists a positive function v on R 2d such that ω(x + y, ξ + η) ω(x, ξ)v(y, η).
An exception concerns [16] by Cordero, Nicola and Rodino, where it is merely assumed that s = σ > 0 and it is evident that in their analysis, ω must be moderate, admitting exponential growth of the symbols.
In [16] it also seems to be the first time where characterizations of symbols satisfying (0.3) in terms of estimates with corresponding short-time Fourier transforms are performed (cf. [16,Theorem 3.1]) and where continuity of operators with infinite order is obtained for Gelfand-Shilov spaces of the forms S s s (R d ) with s less than one (cf. [16,Proposition 4.7]). Here we remark that some implicit steps in such directions are given in [31]. (Cf. Theorems 3.9 and 6.15 in [31].) These continuity properties are established by using methods based on modulation space theory and short-time estimates on the symbols of the operators, instead of the usual micro-local techniques. We also remark that 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 s s (R d ) and Σ s s (R d ) when s ≤ 1. In [9], pseudo-differential operators with symbols satisfying (0.2) with are considered, which for example is interesting in connection with Shubintype pseudo-differential operators. In particular, superexponential growth on the symbols is permitted, giving that the growth conditions on the symbols are even more relaxed compared to [16].
In [9] it is deduced that such operators of infinite order are continuous on the Gelfand-Shilov spaces S s s or Σ s s , depending on the precise conditions on h and r in (0.2), and their distribution spaces. Here it is also proved that such operator classes are algebras under compositions.
In Sect. 3 we extend the results in [9] to the anisotropic case, where the conditions in (0.5) for (0.2) are relaxed into We prove that operators with such symbols are continuous on the (anisotropic) Gelfand-Shilov spaces S σ s (R d ) and Σ σ s (R d ) (again depending on the precise conditions on h and r in (0.2)), and their distribution spaces. We also prove that our operator classes are algebras under compositions, thereby receiving full extension of the results in [9] to the anisotropic case.
In a similar way as in [9,16], our analysis is based on characterizations of our symbols in terms of suitable estimates of their short-time Fourier transforms. On the other hand, an essential part of the analysis in [16] is based on suitable applications of almost diagonalization property for the operators. Such technique works well when ω in (0.2) is moderate. Since this is not the case in our situation when s < 1 or σ < 1, we can not use such approach. Instead we accept certain types of gaps between symbol estimates and estimates on corresponding short-time Fourier transforms, which neither harm our analysis nor threat our conclusions.
Finally we remark that rates of growth and Gevrey regularity are usually different and not so related to each others, leading to differences between the choice of s and the choice of σ. Hence, the restriction s = σ in [9] and in several other contributions or problems, is not natural. We therefore believe that it is relevant to consider, as it is done in Sect. 3, the anisotropic case where s and σ are allowed to be different. An example where anisotropic operators of infinite order appear concerns certain initial value problems for Schrödinger type equations with data in Gelfand-Shilov spaces (see [1]). On the other hand, in the present paper we do not give specific applications which require a long treatise.
The paper is organized as follows. In Sect. 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 Sect. 2 we characterize these symbols in terms of the behavior of their short time Fourier transform. In Sect. 3 we deduce continuity on S σ s (R d ) and Σ σ s (R d ) and their distribution spaces, composition and invariance properties for pseudo-differential operators in our classes. Finally, in order to make it easy for the reader and the community we have, in Appendix A, collected some essential properties and included some short proofs for moderate weights. These properties can essentially be found in the literature, but at different places (see e. g. [20,31]). For example, here we show that moderate weights are bounded by exponential functions.

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.

Gelfand-Shilov Spaces
We start by recalling some facts about Gelfand-Shilov spaces. Let 0 < h, s, σ ∈ R be fixed.  The spaces S σ s (R d ) and Σ σ s (R d ) can be characterized also in terms of the exponential decay of their elements, namely f ∈ S σ s (R d ) (respectively f ∈ Σ σ s (R d )), if and only if for some h, r > 0 (respectively for every h, r > 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. [18] 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 [28] it is proved that for every ε > 0. If s + σ ≥ 1, then the last two inclusions in (1.3) are dense, and if in addition (s, . The Gelfand-Shilov spaces possess several convenient mapping properties. For example they are nuclear and invariant under translations, dilations, and to some extent tensor products and (partial) Fourier transformations, cf. [18,26,27]).
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 (S σ . 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 σ1,σ2 s1,s2 (R d1+d2 ) and Σ σ1,σ2 for some h > 0 respectively 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 [18]. 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 d1 and x 2 ∈ R d2 , 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 d1+d2 ) restrict to homeomorphisms (2) the mappings F 1 and F 2 on S (R d1+d2 ) are uniquely extendable to homeomorphisms . The same holds true if the S σ1,σ2 s1,s2 -spaces and their duals are replaced by corresponding Σ σ1,σ2 s1,s2 -spaces and their duals.

The Short Time Fourier Transform and Gelfand-Shilov Spaces
We recall here some basic facts about the short-time Fourier transform and weights.
Let φ ∈ S σ s (R d )\0 be fixed. Then the short-time Fourier transform of f ∈ (S σ s ) (R d ) is given by Here ( · , · ) L 2 is the unique extension of the The following characterizations of the S σ1,σ2 s1,s2 (R d1+d2 ), Σ σ1,σ2 s1,s2 (R d1+d2 ) and their duals follow by similar arguments as in the proofs of Propositions 2.1 and 2.2 in [32]. The details are left for the reader.
\0 and let f be a Gelfand-Shilov distribution on R d1+d2 . Then the following is true: holds for every r > 0.
Remark 1.5. We notice that any short-time Fourier transform of a Gelfand-Shilov distribution with window function as Gelfand-Shilov function or even a Schwartz function makes sense as a Gelfand-Shilov distribution.
In fact, let (1.9) By defining V φ f as the right-hand side of (1 ). In the same way (1.10) extends uniquely to a continuous map from , then V φ f is still defined as some sort of Gelfand-Shilov distibution, given as the dual of a Gelfand-Shilov space, defined in terms of Komatsu functions (see e. g. [13]).

Weight Functions
) be the set of all weight functions ω on R d1+d2 such that Pseudo-Differential Calculus Page 9 of 33 26 for some r > 0 (for every r > 0). In particular, if ω ∈ P s1,s2 (R d1+d2 ) (ω ∈ P 0 s1,s2 (R d1+d2 )), then for some r > 0 (for every r > 0). The following proposition shows among others limitations concerning growths and decays for moderate weights. (1.14) and The statements in Proposition 1.6 are essentially presented at different places in the literature (cf. [20,30,31]). For conveniency we present a proof in Appendix A, and refer to [20,31] for more facts about weights.

Pseudo-Differential Operators
Let M(d, R) be the set of all d × d-matrices with entries in R, A ∈ M(d, R) and s ≥ 1 2 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 ) is 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. [30]). If instead s, σ > 0 are such that s + σ ≥ 1, a ∈ (S σ,s s,σ ) (R 2d ), then Op A (a) is defined to be the linear and continuous operator from It is easily seen that the latter definition agrees with (1.16) 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.

Symbol Classes
Next we introduce function spaces related to symbol classes of the pseudodifferential operators. These functions should obey various conditions of the form , (1.19) indexed by h > 0.
Definition 1.7. 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 and their topologies are the inductive and the projective topologies of Γ σ,s;h (ω) (R 2d ) respectively, with respect to h > 0.
Furthermore we have the following classes.
, (1.20) where the supremum is taken over all In order to define suitable topologies of the spaces in Definition 1.8, is a Banach space, and the sets in Definition 1.8 are given by and we equip these spaces by suitable mixed inductive and projective limit topologies of (Γ σ1,σ2 s1,s2 ) (h,r) (R d1+d2 ).

Characterizations of Symbols via the Short-Time Fourier Transform
In this section we characterize the symbol class from the previous section in term of estimates of their short-time Fourier transform.
In what follows we let κ be defined as In the sequel we shall frequently use the inequality which follows by straight-forward computations.
, r > 0 and let f be a Gelfand-Shilov distribution on R d . Then the following is true: for every h > 0 (for some h > 0), then f ∈ C ∞ (R d ) and satisfies for every h, r 0 > 0. In particular, for every r 0 > 0. Since |V φ f (x, ξ)| = | F x (ξ)|, the estimate (2.3) 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 , h 2 > 0. Since we get By similar arguments we get the following result. The details are left for the reader.
As a consequence of the previous result we get the following. We also have the following version of Proposition 2.1 , involving certain types of moderate weights.

Proposition 2.5. Let s, σ > 0 be such that
and let a be a Gelfand-Shilov distribution on R 2d . Then the following is true: for some r > 0 (for every r > 0); (2) if (2.10) holds for some r > 0 (for every r > 0), then a ∈ C ∞ (R 2d ) and (2.9) holds for some h > 0 (for every h > 0).
We note that [16, Theorem 3.1] is more general than Proposition 2.5 when s = σ, since the former result is valid for a strictly larger class of window functions. It is also evident that Proposition 2.5 follows from [16, Theorem 3.1] and its proof, also when s and σ are allowed to be different. In order to be self-contained we present a short proof of Proposition 2.5 in Appendix B, where the first part is slightly different compared to the proof of [16, Theorem 3.1].

Invariance, Continuity and Algebraic 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 show 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.

Invariance Properties
An essential part of the study of invariance properties concerns the operator e i AD ξ ,Dx when acting on the symbol classes in the previous sections.
The assertion (1) in the previous theorem is proved in [9] and is essentially a special case of Theorem 32 in [34], whereas (2) can be found in [9,10]. Thus we only need to prove (3) and (4) (1) it follows by straight-forward computation, that the latter condition is invariant under the mapping e i AD ξ ,Dx , and (3) follows from these invariance properties. By similar arguments, taking φ ∈ Σ σ,s s,σ (R 2d ) and using (2) instead of (1), we deduce (4).
We also have the following extension of (4) in [9, Theorem 4.1].
We need some preparation for the proof and start with the following proposition.
By straight-forward application of Leibniz rule in combination with (1.12) we obtain , ξ) , for every fixed h > 0. Let F 2 F a be the partial Fourier transform of F a (x, ξ, y, η) with respect to the (y, η)-variable. Then 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.
for some G 1 satisfying (3.4) in place of G and some c > 0 independent of R. By applying the L ∞ -norm on the last inequality we get We only consider the case q < ∞ when proving the opposite inequality. The case q = ∞ follows by similar arguments and is left for the reader.

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 given in Definitions 1.7 and 1.8. We begin with the case when the symbols belong to Γ σ,s (ω) (R 2d ) or Γ σ,s s,σ;0 (R 2d ). Theorem 3.7. Let A ∈ M(d, R), s, σ > 0 be such that s + σ ≥ 1, ω ∈ P 0 s,σ (R 2d ) and let a ∈ Γ σ,s (ω) (R 2d ). Then Op A (a) is continuous on S σ s (R d ) and on (S σ s ) (R d ).

Theorem 3.8. Let
Remark 3.9. Let s, σ and A be the same as in Theorem 3.8, ω ∈ P 0 s,σ (R 2d ) and let a ∈ Γ σ,s (ω) (R 2d ). Then the following is true: (1) Theorems 3.7 and 3.8 agree in the case when s, σ ≥ 1; (2) Theorem 3.7 is a strict subcase of Theorem 3.8 when s < 1 or σ < 1, because Γ σ,s (ω) (R 2d ) is strictly contained in Γ σ,s s,σ;0 (R 2d ) for such choices s and σ. For example, in this case, there are symbols a which satisfy the hypothesis in Theorem 3.8 and which grow superexponentially in some directions, while the symbols in Theorem 3.7 are allowed to grow at most exponentially, in view of Proposition 1.6; (3) Proposition 4.7 in [16] is a consequence of Theorem 3.7. More precisely, if s = σ ≥ 1 2 and ω = 1, then Theorem 3.7 agrees with Proposition 4.7 in [16], and asserts that Op w (a) is continuous on S s (R d ); (4) the analysis in [16] which lead to Proposition [16,Proposition 4.7], involving a technique on almost diagonalization for pseudo-differential operators can be performed to deduce Theorem 3.7 in the case s = σ ≥ 1 2 . We note that as a corner stone in the analysis in [16], the weight ω needs to be moderate, giving that the symbols in [16] need to be bounded by exponential functions. Consequently, it seems impossible to include symbols with superexponential growth in both x and ξ in the analysis in [16]. In particular, Theorem 3.8 in the case s = σ < 1 seems not possible to reach with the methods in [16].
For the proof of Theorem 3.8 we need the following result. Then and Proof. Since Ω 1 is a bounded set in S σ s;h1 (R d ), there is a constant C > 0 such that 7) 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;2 2+σ h1 (R d ), and the boundedness of Ω 3 in S σ s;h3 (R d ) follows by combining the boundedness of Ω 2 and the boundedness of (3.8) in S σ s;h2 (R d ).  The estimate (3.9) follows from [23], and (3.10) also follows from computations given in e. g. [9,23]. For sake of completeness we present a proof of (3.10).

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 −r0t is attained at t = sk/r 0 , giving that where the last inequality follows from Stirling's formula.
Proof of Theorem 3.8. By Theorem 3.1 it suffices to consider the case A = 0, that is for the operator Observe that Let h 1 > 0 and f ∈ Ω, where Ω is a bounded subset of S σ s,h1 (R d ). For fixed α, β ∈ N d we get By Lemma 3.10 and the fact that (2j)! ≤ 4 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 3.11 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. A ∈ M(d, R), s, σ > 0 be such that s + σ ≥ 1, a ∈ (S σ,s s,σ ) (R 2d ) (a ∈ (Σ σ,s s,σ ) (R 2d )), φ ∈ S σ,s s,σ (R 2d ) (φ ∈ Σ σ,s s,σ (R 2d )), and let and

Lemma 3.12. Let
x − y) be the kernels of Op A (a) and Op A (φ), respectively. Then The essential parts of (3.12) is presented in the proof of [33,Proposition 2.5]. In order to be self-contained we here present a short proof.

Proof. Let
By formal computations, using Fourier's inversion formula we get where all integrals should be interpreted as suitable Fourier transforms.
Before continuing discussing continuity of pseudo-differential operators, we observe that the previous lemma in combination with Propositions 2.3 and 2.4 give the following. A ∈ M(d, R), s, σ > 0 be such that s + σ ≥ 1 and (s, σ) = ( 1 2 , 1 2 ), φ ∈ Σ σ s (R 2d )\0, a be a Gelfand-Shilov distribution on R 2d and let K a,A be the kernel of Op A (a). Then the following conditions are equivalent:
By similar arguments we also get the following result. The details are left for the reader.
Proposition 3.14. Let A ∈ M(d, R), s, σ > 0 be such that s + σ ≥ 1, φ ∈ S σ s (R 2d )\0, a be a Gelfand-Shilov distribution on R 2d and let K a,A be the kernel of Op A (a). Then the following conditions are equivalent: (1) a ∈ Γ σ,s s,σ;0 (R 2d ) (resp. a ∈ Γ σ,s;0 s,σ;0 (R 2d )); Proof. By Theorem 3.1 we may assume that A = 0. Let where h a,x = K a,0 (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 Fubini's theorem on distributions we get Now suppose that r > 0 is arbitrarily chosen. By Proposition 2.3 we get for some c ∈ (0, 1) which depends on s and σ only, that for some r 0 > 0 and with r 1 = (r + r 0 )/c that |J(x, ξ, y, η)| e r0(|x| where r 2 only depends on r and r 0 . Since where h > 0 only depends on r 2 , and thereby depends only on r and r 0 . This implies