Composition operators on Besov and Triebel-Lizorkin spaces with power weights: necessary conditions

Douadi Drihem

Published 2021-08-02Version 1

Let $G:\mathbb{R\rightarrow R}$ be a continuous function. We investigate necessary conditions on $G$ such that \begin{equation*} \{G(f):f\in A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha }) \end{equation*} holds. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ stands for either the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ or the Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$.