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arXiv 2108.00718math.FA
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Besov和Triebel-Lizorkin空间上的幂权复合算子

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

Douadi Drihem

备注：
Here we present the Dahlberg problem on Besov and Triebel-Lizorkin spaces of power weights
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摘要：

设$G:\mathbb{R\rightarrow R}$为连续函数。在对$G$，$s,α,p$和$q$的一些假设下，我们证明了\begin{equation*} \{G(f):f\in A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})\}\subset A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α}) \end{equation*}蕴含着$G$是一个线性函数.这里$A_{p,q}^{s}(\mathbb{R}^{n},|\cdot|^{α})$代表Besov空间$B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$或Triebel-Lizorkin空间$F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$。这些空间统一和推广了许多经典的函数空间，如Sobolev空间的权力。研究这一问题的主要困难之一是具有$α\neq 0$的$A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$空间的范数不是平移不变的，因此必须发展一些新的技术。

英文摘要：

Let $G:\mathbb{R\rightarrow R}$ be a continuous function. Under some assumptions on $G$, $s,α,p$ and $q$ we prove that \begin{equation*} \{G(f):f\in A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})\}\subset A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α}) \end{equation*} implies $G$ is a linear function. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot|^{α})$ stands for either the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$ or the Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$. 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 $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{α})$ spaces with $α\neq 0$ is not translation invariant, so some new techniques must be developed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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