Khavinson problem for hyperbolic harmonic mappings in Hardy space

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 $u={\mathcal{P}}_{\Omega }[\varphi ]$ and $\varphi \in L^p(\partial \Omega ,\mathbb{R})$, where $p\in [1,\infty ]$, ${\mathcal{P}}_{\Omega }[\varphi ]$ denotes the Poisson integral of $\varphi$ 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 $$|\nabla u(x)|\le {\mathbf{C}}_{\Omega ,q}(x)\Vert \varphi {\Vert }_{L^p(\partial \Omega ,\mathbb{R})}$$ and gradient estimate in the direction $l$ $$|⟨\nabla u(x),l⟩|\le {\mathbf{C}}_{\Omega ,q}(x;l)\Vert \varphi {\Vert }_{L^p(\partial \Omega ,\mathbb{R})},$$ 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 $K_0\in \mathbb{N}=\{0,1,2,\dots \}$, 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 $e_n=(0,\dots ,0,1)\in {\mathbb{S}}^{n-1}$. 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 $⟨t_w,w⟩=0$ for $w\in {\mathbb{R}}^n\setminus \{0\}$. \end{abstract}

30 pages