A Beurling-Helson type theorem for modulation spaces

Abstract

We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only 𝒞¹ changes of variables that leave invariant the modulation spaces ℳ^p,q(ℝ^d) are affine functions on ℝ^d. A special case of our result involving the Sjöstrand algebra was considered earlier by A. Boulkhemair.