arXiv2020
\begin{abstract} In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that $u=\mathcal{P}_Ω[ϕ]$ and $ϕ\in L^{p}(\partialΩ, \mathbb{R})$, where $p\in[1,\infty]$, $\mathcal{P}_Ω[ϕ]$ denotes the Poisson integral of $ϕ$ with respect to the hyperbolic Laplacian operator $Δ_{h}$ in $Ω$, and $Ω$ denotes the unit ball $\mathbb{B}^{n}$ or the half-space $\mathbb{H}^{n}$. For any $x\in Ω$ and $l\in \mathbb{S}^{n-1}$, let $\mathbf{C}_{Ω,q}(x)$ and $\mathbf{C}_{Ω,q}(x;l)$ denote the optimal numbers for the gradient estimate $$ |\nabla u(x)|\leq \mathbf{C}_{Ω,q}(x)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})} $$ and gradient estimate in the direction $l$ $$|\langle\nabla u(x),l\rangle|\leq \mathbf{C}_{Ω,q}(x;l)\|ϕ\|_{ L^{p}(\partialΩ, \mathbb{R})}, $$ 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}