Optimal estimates for hyperbolic harmonic mappings in Hardy space

Abstract

Assume that $p\in (1,\infty ]$ and $u=P_h[\varphi ]$, where $\varphi \in L^p({\mathbb{S}}^{n-1},{\mathbb{R}}^n)$. Then for any $x\in {\mathbb{B}}^n$, we obtain the sharp inequalities $$|u(x)|\le \frac{{\mathbf{C}}_q^{\frac{1}{q}}(x)}{(1-|x{|}^2{)}^{\frac{n-1}{p}}}\Vert \varphi {\Vert }_{L^p}\text{and}|u(x)|\le \frac{{\mathbf{C}}_q^{\frac{1}{q}}}{(1-|x{|}^2{)}^{\frac{n-1}{p}}}\Vert \varphi {\Vert }_{L^p}$$ for some function ${\mathbf{C}}_q(x)$ and constant ${\mathbf{C}}_q$ in terms of Gauss hypergeometric and Gamma functions, where $q$ is the conjugate of $p$. This result generalize and extend some known result from harmonic mapping theory ([5, Theorems 1.1 and 1.2] and [1, Proposition 6.16]).

10 pages

References in corpus (1)

• Schwarz lemma for hyperbolic harmonic mappings in the unit ball