Khavinson problem for hyperbolic harmonic mappings in Hardy space

TL;DR

This paper advances the understanding of gradient estimates for hyperbolic harmonic functions in Hardy spaces, partly resolving the generalized Khavinson conjecture by characterizing optimal bounds in various geometric and functional settings.

Contribution

It provides new explicit formulas and conditions for the optimal gradient bounds of hyperbolic harmonic mappings in Hardy spaces, addressing cases previously unresolved.

Findings

Optimal gradient bounds are characterized for different p-norms.

Symmetry properties of the bounds are established in specific geometric domains.

The results partially solve the generalized Khavinson conjecture in hyperbolic harmonic analysis.

Abstract

\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_{\Omega}[\phi]$ and $\phi\in L^{p}(\partial\Omega, \mathbb{R})$, where $p\in[1,\infty]$, $\mathcal{P}_{\Omega}[\phi]$ denotes the Poisson integral of $\phi$ with respect to the hyperbolic Laplacian operator $\Delta_{h}$ in $\Omega$, and $\Omega$ denotes the unit ball $\mathbb{B}^{n}$ or the half-space $\mathbb{H}^{n}$. For any $x\in \Omega$ and $l\in \mathbb{S}^{n-1}$, let $\mathbf{C}_{\Omega,q}(x)$ and $\mathbf{C}_{\Omega,q}(x;l)$ denote the optimal numbers for the gradient estimate $$ |\nabla u(x)|\leq \mathbf{C}_{\Omega,q}(x)\|\phi\|_{ L^{p}(\partial\Omega, \mathbb{R})} $$ and gradient estimate in the direction $l$ $$|\langle\nabla u(x),l\rangle|\leq \mathbf{C}_{\Omega,q}(x;l)\|\phi\|_{ L^{p}(\partial\Omega, \mathbb{R})}, $$ respectively. Here $q$ is the conjugate of $p$. If $q=\infty$ or $q\in[\frac{2K_{0}-1}{n-1}+1,\frac{2K_{0}}{n-1}+1]\cap [1,\infty)$ with $K_{0}\in\mathbb{N}=\{0,1,2,\ldots\}$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;\pm\frac{x}{|x|})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;\pm e_{n})$ for any $x\in \mathbb{H}^{n}$, where $e_{n}=(0,\ldots,0,1)\in\mathbb{S}^{n-1}$. However, if $q\in(1,\frac{n}{n-1})$, then $\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}(x;t_{x})$ for any $x\in\mathbb{B}^{n}\backslash\{0\}$, and $\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;t_{e_{n}})$ for any $x\in \mathbb{H}^{n}$. Here $t_{w}$ denotes any unit vector in $\mathbb{R}^{n}$ such that $\langle t_{w},w\rangle=0$ for $w\in \mathbb{R}^{n}\setminus\{0\}$. \end{abstract}