Weighted Composition Operators Acting on Harmonic Hardy Spaces

TL;DR

This paper characterizes bounded weighted composition operators on harmonic Hardy spaces in higher dimensions and computes their norms for M"obius transformations, advancing understanding of operator behavior in harmonic analysis.

Contribution

It provides a complete characterization of bounded weighted composition operators on harmonic Hardy spaces and explicitly computes their norms for M"obius transformations.

Findings

Characterization of bounded weighted composition operators on $h^p(B)$.

Explicit computation of operator norms for M"obius transformations.

Extension of operator theory to harmonic Hardy spaces in higher dimensions.

Abstract

Suppose $n\geq 3$ and let $B$ be the open unit ball in $\mathbb{R}^n$. Let $\varphi: B\to B$ be a $C^2$ map whose Jacobian does not change sign, and let $\psi$ be a $C^2$ function on $B$. We characterize bounded weighted composition operators $W_{\varphi,\psi}$ acting on harmonic Hardy spaces $h^p(B)$. In addition, we compute the operator norm of $W_{\varphi,\psi}$ on $h^p(B)$ when $\varphi$ is a M\"obius transformation of $B$.