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COMPOSITION OPERATORS ON WIENER AMALGAM SPACES

Part of: Harmonic analysis in several variables Nonlinear operators and their properties

Published online by Cambridge University Press:  08 March 2019

DIVYANG G. BHIMANI
DIVYANG G. BHIMANI*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA email dbhimani@math.umd.edu, divyangbhimani@gmail.com
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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 $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$. 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 in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

MSC classification

Primary: 42B35: Function spaces arising in harmonic analysis 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 42B37: Harmonic analysis and PDE

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Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 257 - 274
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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