Pseudodifferential operators with completely periodic symbols

Motivated by the recent paper of Boggiatto–Garello (J Pseudo-Differ Oper Appl 11:93–117, 2020) where a Gabor operator is regarded as pseudodifferential operator with symbol p(x,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(x,\omega )$$\end{document} periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} quantization.


Introduction.
Consider the formal expression of the Gabor operator: Sf = h,k∈Z d (f, g h,k ) L 2 g h,k , g h,h (t) = e 2πiβk•t g(t − αh), α, β ∈ R + , and that of the pseudodifferential operator with Kohn-Nirenberg quantization: a(x, D)ϕ(x) = e 2πiω•(x−t) a(x, ω)f (t) dt dω, where f ∈ S(R d ), a(x, ω) ∈ S ′ (R 2d ) and the integration is intended in distribution sense.In the recent work [2] it is proven that S = a(x, D), where (1) a(x, ω) = h,k∈Z d e −2πi(x−αh)•(ω−βk) g(x − αh) ḡ(ω − βk), with suitable decay conditions at infinitive of g, ĝ, and convergence in L ∞ (R 2d ).Then using the Calderón -Vaillancourt Theorem about L 2 continuity of pseudodifferential operators, see [18], one can prove that for α + β less than a suitable positive constant C g , depending only on the decay at infinitive of g, ĝ and some of their derivatives, the Gabor operator is invertible as a bounded linear operator on L 2 (R d ).Then as well known in frame theory, the Gabor system {g h,k } h,k∈Z d realizes to be a frame.
The literature about Gabor frame theory is wide, we quote here only the monographies [6], [13], [15], [5].Among others, the problem of finding conditions on the parameters α, β in order to obtain Gabor frames is a challenging one, see for example [10], [14] and the references therein.
Notice now that the symbol in ( 1) is completely periodic, with period α with respect to the spatial variable x and period β with respect to ω.For all the reasons listed above, we think it should be of some relevance to develop the study of pseudodifferential operators with completely periodic symbols, their continuity and possible invertibility in L 2 or more general function spaces.Concerning symbols independent of x, that is Fourier multipliers, we quote the papers [12], [20], where the periodic case in considered.
Wider is the literature concerning the pseudodifferential operators on compact Lie groups, see the fundamental book of Ruzhansky-Turunen [25], which have as particular case symbols periodic in x and discrete (non periodic) in ω.Also interesting is the reversed case where the symbols are discrete in x and periodic in ω; the related operators are called in this case "pseudo-difference" operators, see [3], [19].About pseudodifferential operators on generalized spaces, e.g.modulation spaces, we refer to the following papers [27], [28], [8], [1], [22], [23], [4], [11], [7], [9].The plan of the paper is the following: in §2 we give the notations and definitions; then we review some basic facts about periodic distributions with respect to a general invertible matrix and their Fourier transform.In §3 we introduce the pseudodifferential operators with general τ quantization and for the case of periodic symbols we provide a representation formula, obtained by linear combination of time frequency shift operators.At the end, respectively in §4 and §5 we set the results of continuity and invertibility on general families of time frequency shift invariant spaces.The Appendix A is devoted to give some technicalities in order to compare periodic distributions on R n and distribuitions on the n dimensional torus T n .

Preliminaries
2.1.Notations and basic tools.In whole the paper we will use the following notations and tools: with the well known extension to u ∈ S ′ (R n ).
The polynomial weight function v is defined for some s ≥ 0 by A non negative measurable function m = m(z) on R n is said to be a polynomially moderate (or temperate) weight function if there exists a positive constant C such that For other details about weight functions see [13, §11.1].
In the following we will use in many cases the matrix in GL(2d) which defines the symplectic form, see [13, §9.4],

Time frequency shifts (tfs)
. For z = (x, ω) ∈ R 2d we define the operators: , the next properties easily follow:

Modulation spaces.
Definition 2.1.For a fixed nontrivial function g the short-time Fourier transform (or Gabor transform) of a function f with respect to g is defined as whenever the integral can be considered, also in weak distribution sense.
Definition 2.2.For a fixed g ∈ S(R d ) \ {0} and p, q ∈ [1, +∞], the m−weighted modulation space M p,q m (R d ) consists of all tempered distributions f ∈ S ′ (R d ) such that (with expected modification in the case when at least one among p or q equals +∞).
The definition of the space M p,q m is independent of the choice of the window g, different windows g provide equivalent norms and M p,q m turns out to be a Banach space.In the case of p = q we denote M p m := M p,p m , when m(x, ω) ≡ 1 we write M p,q .For more details about modulation spaces see [13, §6.1, §11].
2.4.Periodic distributions.We say that a distribution u ∈ D ′ (R n ) is periodic (of period 1) if Notice that u is in this case a tempered distribution in S ′ (R n ), then its Fourier transform û can be considered.Moreover it can be shown that where the series converges in S ′ (R n ), (5) c κ (u) := u, φe −2πi •,κ = uφ(κ), For the details see Hörmander [16, §7.2].Now by a straightforward application of Fourier inverse transform we obtain with convergence in S ′ and c κ (u) defined in (5).
Notice that a general periodic distribution u can be regarded as a distribution on the torus T n = R n /Z n , that is a linear continuous form on C ∞ (T n ).In the following D ′ (T n ) will be the topological dual space of C ∞ (T n ).Thus the coefficients in the expansion ( 7) can be regarded as the Fourier coefficients of u, namely In Appendix A we clarify how to make rigorous the above calculations in D ′ (T n ), see in particular (27).For a detailed discussion about distributions on the torus one can see the book of M. Ruzhansky and V. Turunen [25].Consider now and in the whole paper L = (a ij ) ∈ GL(n), the space of invertible matrices of size n × n.
where L −T := (L −1 ) T denotes the transposed of the inverse matrix of L and Thus we obtain for any u ∈ D ′ (T n L ) the Fourier expansion with the Fourier coefficients For short in the following we set holds with convergence in S ′ (R n ), and Here Λ ⊥ := L −T Z n and vol(Λ) := |detL| = meas (L[0, 1] n ) are respectively called dual lattice and volume of Λ.

Pseudodifferential Operators with periodic symbol
We say τ pseudodifferential operator, 0 The formal integration must be understood in distribution sense.For the definition and development of pseudodifferential operators see the basic texts [26], [17].
For I and 0 respectively the identity and null matrices of dimension d × d, let us introduce the d × 2d matrices Then for any 0 ≤ τ ≤ 1 and u ∈ S(R d ) we can write where c κ (p) are the Fourier coefficients defined in (10) and J the matrix introduced in (3).
Proof.Using ( 9), (10) we perform the Fourier expansion of the symbol Considering the decomposition we obtain Let us set for simplicity of notation L −T j = I j L −T , j = 1, 2. Then in view of convergence in S ′ and formal integration in distribution sense it follows The proof ends by observing that, thanks to (4), In view of (14) the pseudodifferential operator Op τ may be written in lattice notation with A, B diagonal matrices defined as in (13), it is trivial to show that Thus for any 0 ≤ τ ≤ 1 we have with convergence in S ′ (R d ) and c h,k (p) defined by formal integration

Continuity
We say that a Banach space S(R d ) ֒→ X ֒→ S ′ (R d ), with S(R d ) dense in X, is time frequency shifts invariant (tfs invariant from now on) if for some polynomial weight function v and C > 0 Example 3.
For z = (x, ω) ∈ R 2d , assuming (x 0 , ω 0 ) = (0, 0) and using (2), we compute: Then for any τ ∈ [0, 1] the operator Op τ (p) is bounded on X and In lattice terms, see (15), we can write Proof.Using Proposition 3.1 and in view of the tfs invariance ( 16) we obtain for where C is the constant in (16).The proof follows from the density of S(R d ) in X.

4.1.
The case of Fourier Multipliers.Assume now that the symbol is independent of x, namely consider a Fourier multiplier σ = σ(ω) ∈ S ′ (R d ), P −periodic, with P ∈ GL(d), that is In such a case the related pseudodifferential operator, as a linear bounded operator from S(R d ) to S ′ (R d ), reads as Inserting within (18) the Fourier expansion of σ where the series is convergent in with convergence of the series in S ′ (R d ); in (18), (19), c k (σ) stand as usual for the Fourier coefficients of σ.
The following result shows that in the case of Fourier multiplier operators the sufficient boundedness condition given in Theorem 4.1 is also necessary for σ(D) to be extended as a linear bounded operator in the weighted Lebesgue space . The function T −P T k ũ will be then supported on P k := P 0 − P −T k for all k ∈ Z d .Since the set collection {P k } k∈Z d defines a covering on R d such that P k ∩ P h has zero Lebesgue measure, whenever k = h, we get It is even clear that σ(D)ũ reduces to c k (σ)T −P −T k ũ in the interior of the set P k , for each k, so that by change of integration variable and sub-multiplicativity of v, we get with K > 0 depending only on P and v.
Remark 3. Combining Proposition 4.1 and Theorem 4.1 we obtain that, in the case of Fourier multipliers, condition (21) is actually equivalent to the continuity in any tfs invariant Banach space, as defined in (16); here translation invariance of the space is enough, due to the Fourier multiplier structure (20).Instead, condition (17) is no longer necessary for continuity of periodic pseudodifferential operators, with x-dependent symbol, in tfs invariant spaces.It can be easily shown by taking a symbol of the following form p(x, ω) = ν(x)σ(ω) where ν is a function in L ∞ (T), such that the sequence of its Fourier coefficients {c k (ν)} / ∈ ℓ 1 (Z) and σ(ω) satisfies (21).For instance we could take ν(x) = 1 for 0 ≤ x < 1/2, ν(x) = 0 for 1/2 ≤ x < 1, repeated by periodicity.It is straightforward to check that the pseudodifferential operator p(x, D) maps continuously L p v (R) into itself, whenever 1 ≤ p < +∞.On the other hand we compute at once that 1 Such a function u can be defined by ũ(x) := |detP | , where ψ is any non negative smooth function supported on the unit cube Q = [0, 1] d such that ψ(z)dz = 1.

Invertibility
For the study of invertibility condition of pseudodifferential operators we will make use of the well known properties of the von Neumann series in Banach algebras, see e.g.[24], in the following version.Proposition 5.1.Consider x ∈ A, where (A, • ) is a Banach algebra on the field of complex numbers, with multiplicative identity e.If there exists c ∈ C \ {0} such that e − cx < 1, then x is invertible in A and c 0 (p) = 0 and where C is the constant in (16).Then for any 0 ≤ τ ≤ 1 i) the operator Op τ (p) is invertible in L(X); ii) the norm in L(X) of the inverse operator satisfies the following estimate Notice that, according to the previous estimate, the invertibility of Op τ (p) is independent of the quantization τ .
where C is the constant in (16).Thus i) directly follows from Proposition 5.1 .Thanks to the assumption (22), the estimate (23) and Proposition 5.1, the inverse operator (Op τ (p)) −1 can be expanded in Neumann series then using again (23) we have , which proves ii).
Appendix A. On periodic distributions and distributions on the torus This section is devoted to shortly review some known facts about the comparison between the space of periodic distributions in R n , see Section 2.4, and the space D ′ (T n L ) of distributions on the torus T n L := R n /LZ n , in order to justify the identification of the aforementioned spaces, that we have implicitly assumed in the whole paper.Recall that D ′ (T n L ) is the space linear continuous forms on the function space C ∞ (T n L ) (the latter being endowed with its natural Fréchet space topology).As already mentioned in Section 2.4, the reader is referred to Ruzhansky -Turunen [25] for a thorough study of distributions on the torus.For the rest, the results collected herebelow come essentially from making explicit some of the results established in Hörmander [16,Section 7.2].It is well understood that functions on the torus T n L can be naturally identified with L−periodic functions in R n .Below, we will illustrate a way to extend the same identifications to all L−periodic distributions in R n .This extension to distributions can be made by a duality argument, as it is customary.Thought the following arguments work in the case of L−periodicity, with arbitrary invertible matrix L ∈ GL(n), just for simplicity we will restrict to the case of L = I n the n × n identity matrix, leading to 1−periodic functions and distributions.So let us first consider a 1−periodic measurable function f On the other hand, when identified (as usual) with an integrable function on the torus T n , f defines an element of D ′ (T n ), whose action on test functions in order to avoid confusion, here and below •, • T n stands for the duality pair between D ′ (T n ) and C ∞ (T n ), whereas •, • is denoting the dual pair between S(R n ) and S ′ (R n ).
Testing f against an arbitrary function ϕ ∈ S(R n ) we compute where the countable-additivity of the Lebesgue integral, together with Fubini's theorem to interchange the sum and the integral, and the periodicity of f are used.The function which appears in the last line in (24), is a 1−periodic C ∞ function that can be canonically identified with a (unique) element in C ∞ (T n ) ); we call it 1−periodization of ϕ.It is fairly easy to check that the mapping ϕ → ϕ per is continuous as a ), as well).Therefore the calculations above show naturally the way to define a linear mapping T from the space D ′ (T n ) to the subspace of D ′ (R n ) consisting of 1−periodic distributions (which are automatically tempered distributions), just by setting for It is easy to verify that T acts continuously from D ′ (T n ) to 1−periodic distributions in R n .Moreover (24) shows that T (U ) properly reduces to the 1−periodic tempered distribution in R n corresponding to a function U ∈ L 1 (T n ).
It is a little less obvious that T is invertible, so that it actually defines an isomorphism.This can be proved by noticing that every function ψ ∈ C ∞ (T n ) can be regarded as the 1−periodization of (at least) one function ϕ ∈ S(R n ), namely ψ = ϕ per .To see this, consider a function φ ∈ C ∞ 0 (R n ) satisfying (6) and set (26) ϕ := φψ , for any function ψ ∈ C ∞ (T n ) (identified with its 1−periodic C ∞ −counterpart in R n ).Of course, ϕ defined above belongs to C ∞ 0 (R n ) so it is rapidly decreasing in R n ; moreover, in view of ( 6) and the periodicity of ψ, we get for any x ∈ R n ϕ per (x) = showing that ψ is actually the 1−periodization of ϕ.This leads to associate to any periodic distribution u ∈ S ′ (R n ) a linear form U on C ∞ (T n ) by setting for every ψ ∈ C ∞ (T n ) (27) U, ψ T n := u, ϕ , being ϕ = ϕ(x) the rapidly decreasing (actually compactly supported smooth) function in R n associated to ψ as in (26).In order to give consistency to the definition of U , we must prove that it is independent of ϕ.Let us notice that, thanks to (5), any function ϕ ∈ S(R n ), whose 1−periodization is given by ψ ∈ C ∞ (T n ), satisfies the following: φ(κ) = c κ (ψ) , ∀ κ ∈ Z n .Thus using the Fourier expansion (7) we obtain u, ϕ = This shows the consistency of the definition of U above; the continuity of the linear form U on C ∞ (T n ) also easily follows, so that U ∈ D ′ (T n ).The mapping u → U defined on 1−periodic distributions in R n by (27) provides a linear continuous operator from the space of 1−periodic (tempered) distributions in R n to D ′ (T n ), which is actually the inverse of the operator T introduced before, see (25).This equivalently shows that T is an isomorphism, up to which 1−periodic distributions in R n can be thought to be elements of D ′ (T n ) and viceversa.Thus a 1−periodic distribution u ∈ S ′ (R n ) can be regarded as a linear continuous form on C ∞ (T n ); in particular this makes rigorous the testing of u against 1−periodic smooth functions in C ∞ (T n ), such as e −2πi κ,• , with κ ∈ Z n , providing meanwhile an explicit explanation of formula (8) for the Fourier coefficients of u.

( 15 )Example 2 .
Op τ (p) = µ∈Λ ⊥ p(µ)e 2πiτ I2µ,I1µ π J µ .Let a = (a 1 , . . .a d ), b = (b 1 , . . ., b d ) be two vectors in (R \ {0}) d .Using the notation in Example 1, we say that a symbol p ∈ S ′ (R 2d ) is ab-periodic if, for any κ = (h, k) ∈ Z 2d , p(• + ah, • + bk) = p(•, •).Considering the matrix 3.5].• The m−weighted Lebesgue space L p m (R d ) and m−weighted Fourier-Lebesgue space F L p m (R d ) are respectively defined as the sets of measurable functions and tempered distributions in R d , making finite the norms f L p m ) and F L P m (R d ) are time frequency shifts invariant for any p ∈ [1, +∞].In both the examples the positive constants C are directly obtained by (2) and depend only on the weights m.