A Beurling-Helson type theorem for modulation spaces

We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only $\mathcal{C}^{1}$ changes of variables that leave invariant the modulation spaces $\M{p,q}(\rd)$ are affine functions on $\rd$. A special case of our result involving the Sj\"ostrand algebra was considered earlier by A. Boulkhemair.


Introduction
Given a function φ defined from the torus T to itself, let φ * be the change of variables defined by (1) φ * (u) = u • φ for any function u defined on T.
In 1953, A. Beurling and H. Helson proved that if φ is continuous from T into itself and if φ * is a bounded linear operator on the Fourier algebra A(T) = A 1 (T) of absolutely convergent Fourier series, then necessarily φ(t) = kt + φ(0) for some k ∈ Z [1]. The proof of this result involved some nontrivial arithmetical considerations. A different proof was given by J.-P. Kahane [15]. The Beurling-Helson theorem was later extended to the higher dimensional setting by W. M. Self [17]. More recently, V. Lebedev and A. Olevskiǐ [16] further extended and generalized the Beurling-Helson theorem. In particular, for d ≥ 1 and 1 ≤ p < ∞ let A p (R d ) = F L p (R d ) equipped with the norm f Ap = f L p where F is the Fourier transform defined by F f (ω) = f (ω) = R d f (t) e −2πit·ω dt. It was proved in [16] that if φ : R d → R d is C 1 , and if φ * maps A p (R d ) into itself for some 1 ≤ p < ∞, p = 2, then φ(x) = Ax + φ(0) where A is a real invertible d × d matrix. In this higher dimensional setting, the case p = 1 was already proved in [17]. Observe that since for any homeomorphism φ on T such that φ −1 satisfies the Lipschitz condition, φ * is bounded on A 2 (T), and a transference argument can be used to prove similar result for A 2 (R d ).
In this note, we shall characterize the C 1 changes of variables that leave invariant the modulation spaces (to be defined below). In particular, our result applies to a special subspace of the Fourier algebra called Feichtinger algebra. This space denoted S 0 was introduced by H. Feichtinger [5] and is the smallest Banach algebra that is invariant under both the translation and the modulation operators. Moreover, the Feichtinger algebra is an example of a modulation space and plays an important role in the theory Gabor frames [11]. In fact, the modulation spaces have also been playing an increasing role in the analysis of pseudodifferential operators [12,13,20]. Furthermore, a Banach algebra of pseudodifferential operators known as the Sjöstrand algebra, denoted S ω , and which contains the Hörmander class S 0 0,0 , was introduced independently by Feichtinger [6] and J. Sjöstrand [18]. This space is yet another example of a modulation space. We refer to [7] for an updated version of [6] which contains some historical perspectives on the modulation spaces. In 1997, A. Boulkhemair [4] proved that if φ is a C 1 mapping on R d such that φ * maps S ω into itself, then φ must be an affine function: This is a Beurling-Helson type theorem for the Sjöstrand algebra. It is therefore natural to seek a characterization of the changes of variables that leave invariant modulation spaces. The goal of this note is to extend and generalize this Beurling-Helson type theorem to all the modulation spaces. The main argument in the proof of our result is the fact that the intersection of a modulation space with the space of functions with compact support coincides with the subspace of compactly supported functions in A p (R d ) = F L p (R d ). The proof of this fact as well as the definition of the modulation spaces will be given in Section 2. Finally in Section 3 we shall prove our main result. In the sequel, we shall denote by |A| the Lebesgue measure of a measurable subset A of R d .

Preliminaries
whenever the integral makes sense. This definition can be extended to f ∈ S ′ (R d ) and g ∈ S(R d ) and yields a continuous function V g f , see [11]. Definition 1. Given 1 ≤ p, q ≤ ∞, and given a window function 0 = g ∈ S, the modulation space M p,q = M p,q (R d ) is the space of all distributions f ∈ S ′ for which the following norm is finite: with the usual modifications if p and/or q are infinite.
Remark 1. The definition is independent of the choice of the window g in the sense of equivalent norms. The modulation spaces were originally introduced by Feichtinger [6]. We refer to [11] and the references therein for more details about modulation spaces. The Feichtinger algebra S 0 which coincides with the modulation space M 1,1 (R d ) is a Banach algebra under both pointwise multiplication and convolution. Furthermore, [5,8,9].
While the Beurling-Helson theorem completely classifies the changes of variables that operate in A 1 (T) (and also on A 1 (R)) it was still unknown what changes of variables operate on the Feichtinger algebra and more generally on the modulation spaces. This question will be completely settled below.

2.2.
Local modulation spaces. The theory of modulation can be defined in the general setting of locally compact Abelian groups [7]. In particular, it can be shown that for G = Z d (or any discrete group), Here we focus on functions that are locally in a modulation space.
In the sequel we shall denote by M p,q comp (R d ) the subspace of M p,q (R d ) consisting of compactly supported functions, and by M p,q loc (R d ) the space of functions that are The next result contains the key argument in the proof of our main result. We wish to point out that some special cases of the result are already known. For instance, the result was proved for M ∞,1 (R d ) in [4, Theorem 5.1], while [9] dealt with M p,p (R d ) 1 ≤ p < ∞. Furthermore, an independent and different proof of part b. of Lemma 1 using convolution relations on generalized amalgam spaces was indicated to us by H. Feichtinger [10].
Proof. We shall only prove part a. of the result as part b. follows from the definition of M p,q loc (R d ). Furthermore, to prove a. it suffices to show that given a compact subset Note that this last equation holds not only as set equality, but also as equality of Banach spaces with equivalent norms.
Let R > 0 be given and let For the converse, let R > 0 be given and u ∈ M p,q (R d ) such that supp(u) ⊂ B R (0). Let g ∈ C ∞ 0 (R d ) such that g ≡ 1 on B 2R (0). It is trivially seen that for all x ∈ B R (0) and for all t ∈ B R (0), g(t − x) = 1. Thus, for all ω ∈ R d and for x ∈ B R (0), Therefore,

Main results
Before stating our main result, we wish to indicate that it is trivially seen that all the modulation spaces are invariant under affine changes of variables. That is, let 1 ≤ p, q ≤ ∞ and φ : R d → R d be an affine mapping, i.e., φ(x) = Ax + b where A is a d × d real invertible matrix and b ∈ R d . Then the linear operator φ * given by (1) If we restrict our attention to the modulation spaces M p,p (R d ) 1 < p < ∞ the following stronger result can be proved. For Proposition 1 we assume that R d = ∪ N k=1 Q k where for each k, Q k is a (possible infinite) "cube" with sides parallel to the coordinates axis. Moreover, we assume that for k = 1, . . . , N the Q k s have disjoint interiors. Proposition 1. Let φ be a continuous on R d such that for k = 1, . . . , N, the restriction φ k of φ to Q k is an affine function given by φ k (x) = A k x + b k where A k is a real invertible d × d matrix and b k ∈ R d . Then the linear operator φ * given by (1) maps Proof. It is evident from the definition of the modulation spaces that M p,p is invariant under the Fourier transform, see [8,9]. Let u ∈ M p,p (R d ), then Moreover, note that χ Q k is a bounded Fourier multiplier on all M p,p (R d ): this follows from [2, Theorem 1] in the case d = 1, and from [3, Theorem 6] when d > 1. Consequently, using the invariance of M p,p (R d ) under the Fourier transform, we conclude that there exists c k > 0 such that from which the proof follows.
Remark 2. The conclusion of Proposition 1 holds if we used an infinite decomposition of R d , that is if we assume that R d = ∪ ∞ k=1 Q k where the cubes Q k still have sides parallel to the coordinate axis and disjoint interiors. In this case, the extra assumption needed to prove the previous result is that the constants c k appearing in the above proof, are uniformly bounded, i.e., sup k c k < ∞.
We are now ready to state and prove our main result.
In particular, the Feichtinger algebra M 1,1 (R d ) is preserved by, and only by affine changes of variables.
Remark 4. Using Lemma 1 and [16, pp. 214], it follows that if φ : R d → R d is nonlinear and C 2 , then φ * is not bounded on M p,q . This fact together with Proposition 1, show that the C 1 condition in Theorem 1 is the only nontrivial smoothness condition to impose on φ.
For the Sjöstrand algebra S ω which coincides with the modulation space M ∞,1 (R d ), Theorem 1 was proved in under a weaker assumption on φ. More specifically, it was proved in [4, Theorem 5.1] that if φ is a proper mapping, i.e., φ is continuous on R d and φ −1 (K) is a compact set for any compact subset K of R d , and if φ * (M ∞,1 (R d )) ⊂ M ∞,1 (R d ) then φ(x) = Ax+φ(0). It is also straightforward to prove Theorem 1 under this weaker assumption on φ.
Finally, we wish to conclude this paper by pointing out the connection of our main result to certain Fourier multipliers. More precisely, let σ be a function defined on R d . The Fourier multiplier with symbol σ is the operator H σ initially defined on S by We refer to [19] for more on Fourier multipliers. As mentioned above, there is a strong connection between the L p -continuity of the Fourier multipliers and the Beurling-Helson theorem. In particular, the family of homomorphisms e iφ(ξ) on the space of L p -Fourier multipliers was investigated by Hörmander in [14,Section 1.3]. It is easily seen that σ 0 (ξ) = e iξ , then H σ 0 is bounded on all L p (R d ) for 1 ≤ p ≤ ∞ and d ≥ 1. Hörmander proved that if φ : R d → R d is C 2 and if φ * (σ 0 )(ξ) = σ 0 (φ(ξ)) = e iφ(ξ) is a bounded Fourier multiplier on L p (R d ) for some 1 < p < ∞ and p = 2, then φ is an affine function [14,Theorem 1.15]. It is interesting to note that there exist nonlinear (non-affine) functions φ on R d such that the Fourier multipliers with symbols φ * (σ 0 )(ξ) = σ 0 (φ(ξ)) = e iφ(ξ) are bounded on all modulation spaces [3].

Acknowledgment
The author would like to thank Chris Heil for bringing some of the questions discussed in this work to his attention. He also thanksÁrpád Bényi, Hans Feichtinger, Karlheinz Gröchenig, Norbert Kaiblinger, and Luke Rogers for helpful discussions.