Schwarz lemma for harmonic mappings in the unit ball

TL;DR

This paper generalizes the Schwarz lemma for harmonic mappings in the unit ball, establishing bounds involving $L^p$ norms and providing sharp constants for the derivative at the origin.

Contribution

It introduces a generalized Schwarz lemma for harmonic mappings with $L^p$ norm bounds and determines sharp constants for the derivative at zero.

Findings

Bound on harmonic mappings: $|u(x)| \\le g_p(|x|) \\|u\|_p$

Sharp constant $C_p$ for the derivative at zero

Extension of classical harmonic mapping results

Abstract

We prove the following generalization of Schwarz lemma for harmonic mappings. If $u$ is a harmonic mapping of the unit ball $B_n$ onto itself such that $u(0)=0$ and $\|u\|_p:=\left(\int_S|u(\eta)|^pd\sigma(\eta)\right)^{1/p}<\infty$, $p\ge 1$ then $|u(x)|\le g_p(|x|)\|u\|_p$ for some smooth sharp function $g_p$ vanishing in $0$. Moreover we provide sharp constant $C_p$ in the inequality $\|Du(0)\|\le C_p\|u\|_p$. Those two results extend some known result from harmonic mapping theory (\cite[Chapter~VI]{ABR}).