A Beurling-Helson type theorem for modulation spaces

TL;DR

This paper establishes a Beurling-Helson type theorem for modulation spaces, showing that only affine transformations preserve these spaces among smooth changes of variables, extending previous results involving the Sjöstrand algebra.

Contribution

The paper proves a new invariance theorem for modulation spaces under smooth changes of variables, specifically characterizing affine functions as the only invariants.

Findings

Only affine functions leave modulation spaces invariant under $ ext{C}^1$ changes of variables.

Extension of previous results involving the Sjöstrand algebra.

Provides a characterization of invariance properties of modulation spaces.

Abstract

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.