Nonlinear operations on a class of modulation spaces

TL;DR

This paper investigates the conditions under which nonlinear functions map modulation spaces into themselves, extending known results from analytic to certain non-analytic functions for specific parameter ranges.

Contribution

It extends the understanding of nonlinear operations on modulation spaces, showing that non-analytic functions preserve the space under certain conditions, broadening previous results limited to analytic functions.

Findings

Modulation spaces are closed under nonlinear operations for certain parameters.

The space is a multiplication algebra when s > n - n/q.

Non-analytic functions can also map the space into itself if q ≥ 4/3.

Abstract

We discuss when the nonlinear operation $f\mapsto F(f)$ maps the modulation space $M^{p,q}_s(\mathbb{R}^n)$ ($1 \leq p,q \leq \infty$) to the same space again. It is known that $M^{p,q}_s(\mathbb{R}^n)$ is a multiplication algebra when $s > n-n/q$, hence it is true for this space if $F$ is entire. We claim that it is still true for non-analytic $F$ when $q\geq4/3$.