$H^p$-Norm estimates of the partial derivatives and Schwarz lemma for $\alpha$-harmonic functions

TL;DR

This paper studies the properties of $ abla f$ for $eta$-harmonic functions, establishing conditions for membership in generalized Hardy spaces and deriving a Schwarz lemma, thus extending classical harmonic analysis results.

Contribution

It provides new criteria for the membership of derivatives of $eta$-harmonic functions in generalized Hardy spaces and introduces a Schwarz lemma for these functions.

Findings

Both $f_z$ and $f_{ar{z}}$ are in $ ext{H}_ ext{G}^p( ext{D})$ for $eta>0$.

Derivatives are in $ ext{H}_ ext{G}^p( ext{D})$ iff $f$ is analytic when $eta<0$.

A Schwarz lemma for $eta$-harmonic functions is established.

Abstract

Suppose $\alpha>-1$ and $1\leq p \leq \infty$. Let $f=P_{\alpha}[F]$ be an $\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2\pi)$, where $\dot{F}(e^{i\theta}):=\frac{dF(e^{i\theta})}{d\theta}$. In this paper, we investigate the membership of $f_z$ and $f_{\overline{z}}$ in the space $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$, the generalized Hardy space. We prove, if $\alpha>0$, then both $f_z$ and $f_{\overline{z}}$ are in $\mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$. If $\alpha<0$, then $f_z$ and $f_{\overline{z}}\in \mathcal{H}_{\mathcal{G}}^{p}(\mathbb{D})$ if and only if $f$ is analytic. Finally, we investigate a Schwartz Lemma for $\alpha$-harmonic functions.