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

Abstract

Suppose that $1<p\le \infty$ and $\ddot{I}†\in L^p({\mathbb{B}}^n,{\mathbb{R}}^n).$ In this note, we use Hölder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_p$ and function $C_p(x)$ in the following inequalities $$|u(x)|\le \frac{C_p}{(1-|x{|}^2{)}^{(n-1)/p}}\cdot ||\ddot{I}†|{|}_{L^p}$$ and $$|u(x)|\le \frac{C_p(x)}{(1-|x{|}^2{)}^{(n-1)/p}}\cdot ||\ddot{I}†|{|}_{L^p},$$ 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}).

The extremum function $\ddot{I}{†}_0(eta)$ given in arXiv: 2312.15879 is incorrect. In this new version, we have rephrased the main theorems and obtained the correct expression for the extreme value function $\ddot{I}{†}_0(\hat{I}\cdot )=\left(0,0,\dots ,{\left[\frac{(1-|x{|}^2{)}^{{\hat{I}}^2-\frac{n-1}{q}}}{|x-\hat{I}\cdot {|}^2}\right]}^{q/p}\right)$