Khavinson Problem for Hyperbolic Harmonic Mappings in Hardy Space

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