Composition Operators on Wiener amalgam Spaces

TL;DR

This paper characterizes when composition operators map Wiener amalgam spaces to different Wiener spaces, showing that such operators are real analytic functions with specific properties, and extends results to modulation spaces, answering an open question.

Contribution

It establishes necessary and sufficient conditions for composition operators on Wiener amalgam spaces, linking their boundedness to the real analyticity of the composing function, and extends these results to modulation spaces.

Findings

Characterization of composition operators on Wiener amalgam spaces.

Extension of results to modulation spaces.

Resolution of an open question by Bhimani-Ratnakumar.

Abstract

For a complex function $F$ on $\mathbb C$, we study the associated composition operator $T_{F}(f):=F\circ f= F(f)$ on Wiener amalgam $W^{p,q}(\mathbb R^d) \ (1\leq p< \infty, 1\leq q<2).$ We have shown $T_{F} $ maps $W^{p, 1}(\mathbb R^d)$ to $W^{p,q}(\mathbb R^d)$ if and only if $F$ is real analytic on $\mathbb R^2$ and $F(0)=0.$ Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb R^d).$ In particular, this gives an affirmative answer to the open question proposed by Bhimani-Ratnakumar (J. Funct. Anal. 270 (2016), p.621-648).