Composition Operators on Wiener amalgam Spaces

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 $W^{p,q}({\mathbb{R}}^d)(1\le p<\infty ,1\le q<2).$ 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).

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