Khavinson's conjecture for hyperbolic harmonic functions on the unit ball

Let $n\geq 2$ and let ${\mathbb B}^n$ and ${\mathbb S}^{n-1}$ denote the unit ball and unit sphere in ${\mathbb R}^n$. For $p\in(1,\infty]$, let $q$ be its conjugate, let $\phi\in L^p({\mathbb S}^{n-1},{\mathbb R})$, and let $u=\mathcal{P}_h[\phi]$ be the invariant Poisson integral, where

$$\mathcal{P}_h[\phi](x)=\int_{{\mathbb S}^{n-1}}\mathcal{P}_h(x,\zeta)\phi(\zeta)\,d\sigma(\zeta),$$

and

$$\mathcal{P}_h(x,\zeta)=\left(\frac{1-|x|^2}{|x-\zeta|^2}\right)^{n-1}.$$

For $x\in{\mathbb B}^n\setminus\{0\}$ and $\ell\in{\mathbb S}^{n-1}$, let $\mathcal{C}_p(x)$ and $\mathcal{C}_p(x;\ell)$ be the optimal constants in

$$|\nabla u(x)|\leq\mathcal{C}_p(x)\|\phi\|_p,$$

and

$$|\langle\nabla u(x),\ell\rangle|\leq\mathcal{C}_p(x;\ell)\|\phi\|_p.$$

Khavinson's conjecture. Let $p\in(1,\infty]$, $n\geq3$, and $x\in{\mathbb B}^n\setminus\{0\}$. Then

$$\mathcal{C}_p(x)= \begin{cases} \mathcal{C}_p(x;n_x),&1<p<n,\\ \mathcal{C}_p(x;t_x),&p>n, \end{cases}$$

where $n_x=\frac{x}{|x|}$ and $t_x$ is any unit vector satisfying $\langle t_x,x\rangle=0$. Moreover, if $p=n$ or $p=\infty$, then $\mathcal{C}_p(x;\ell)$ does not depend on $\ell$.

This conjecture identifies the direction giving the sharp gradient estimate for hyperbolic harmonic Poisson integrals. The planar case $n=2$ was solved by Kalaj and Marković, and the general conjecture is reported in the source as recently proved by Liu; consequently, the conjecture is solved.