On the composition operators on Besov and Triebel-Lizorkin spaces of power weights

TL;DR

This paper investigates the conditions under which composition operators preserve Besov and Triebel-Lizorkin spaces with power weights, showing that such operators are linear functions under certain assumptions.

Contribution

It proves that composition operators mapping these weighted function spaces into themselves are necessarily linear, extending classical results to weighted Besov and Triebel-Lizorkin spaces.

Findings

Composition operators are linear under certain conditions.

Weighted spaces with power weights are preserved only by linear functions.

New techniques are developed to handle non-translation-invariant norms.

Abstract

Let $G:\mathbb{R\rightarrow R}$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove 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*} implies $G$ is a linear function. 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 $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$. These spaces unify and generalize many classical function spaces such as Sobolev spaces of power weights. One of the main difficulties to study this problem is that the norm of the $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ spaces with $\alpha \neq 0$ is not translation invariant, so some new techniques must be developed.