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

TL;DR

This paper addresses sharp, $L^{p}$-type bounds for mappings on the unit ball that admit general Poisson representations $u_{\alpha,\beta}$ with kernel $P_{\alpha,\beta}$. It combines Hölder's inequality with basic hypergeometric function properties to derive explicit sharp constants $C_{p}$ and $C_{p}(x)$ in the pointwise bound $|u_{\alpha,\beta}[\varphi](x)| \le \frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}} \|\varphi\|_{L^{p}}$ (and the uniform version) for $1<p\le\infty$, under $\beta-\alpha\ge n-1$ and $\alpha\ge 1$. The main results express these constants via hypergeometric functions and gamma functions, and are shown to be sharp through explicit extremal test functions; the framework extends known estimates for harmonic and hyperbolic-harmonic mappings. The work thus provides optimal $L^{p}$ control of general Poisson integrals in the unit ball with broad applicability to related Dirichlet problems and geometric function theory.

Abstract

Suppose that $1<p\leq\infty$ and $\varphi\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ 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 $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||\varphi||_{L^{p}}$$ 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}).

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1
• Remark 2
• Corollary 1.5
• Lemma 2.1
• Lemma 2.2
• proof