Optimal estimates for mappings admitting general Poisson representations in the unit ball

Abstract

Suppose that $1<p\leq\infty$ and $\varphi\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use H\"{o}lder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_{p}$ and function $C_{p}(x)$ in the following inequalities $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||\varphi||_{L^{p}}$$ and $$|u(x)|\leq \frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}}\cdot||\varphi||_{L^{p}},$$ where $u$ are those mapping from the unit ball $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$ admitting general Poisson representations. The obtained results generalize and extend some known results from harmonic mappings (\cite[Proposition 6.16]{ABR92} and \cite[Theorems 1.1 and 1.2]{DM12}) and hyperbolic harmonic mappings (\cite[Theorems 1.1 and 1.2]{CJLK20}).