\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}
Categories: math.AP math.CA