Khavinson Problem for Hyperbolic Harmonic Mappings in Hardy Space

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 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 ∈Ω and $l\in \mathbb {S}^{n-1}$, let C_Ω,q(x) and C_Ω,q(x;l) denote the optimal numbers for the gradient estimate

$ \left |\nabla u(x)\right |\leq \mathbf {C}_{\Omega ,q}(x)\left \|\phi \right \|{\!}_{L^{p}(\partial {\Omega }, \mathbb {R})} $

and the gradient estimate in the direction l

$ \left |\langle \nabla u(x),l\rangle \right |\leq \mathbf {C}_{\Omega ,q}(x;l)\left \|\phi \right \|{\!}_{L^{p}(\partial {\Omega }, \mathbb {R})}, $

respectively. Here q is the conjugate of p. If $q\in [1,\infty ]$, then $\mathbf {C}_{\mathbb {B}^{n},q}(0)\equiv \mathbf {C}_{\mathbb {B}^{n},q}(0;l)$ for any $l\in \mathbb {S}^{n-1}$. If $q=\infty$, q = 1 or $q\in [\frac {2K_{0}-1}{n-1}+1,\frac {2K_{0}}{n-1}+1]$ with $K_{0}\in \mathbb {N}$, 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 $x\in \mathbb {H}^{n}$. 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 $w\in \mathbb {R}^{n}\setminus \{0\}$.