Schwarz lemma for harmonic mappings in the unit ball

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}).