The generalized Khavinson conjecture for hyperbolic harmonic mappings

Let $p\in[1,\infty]$ and let $q$ be its conjugate. For $\Omega=\mathbb{B}^{n}$ or $\mathbb{H}^{n}$, suppose that $u=\mathcal{P}_{\Omega}[\phi]$ with $\phi\in L^{p}(\partial\Omega,\mathbb{R})$. For $x\in\Omega$ and $l\in\mathbb{S}^{n-1}$, let $\mathbf{C}_{\Omega,q}(x)$ and $\mathbf{C}_{\Omega,q}(x;l)$ be the optimal constants in

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

and

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

Thus $\mathbf{C}_{\Omega,q}(x)=\sup_{l\in\mathbb{S}^{n-1}}\mathbf{C}_{\Omega,q}(x;l)$. Let $e_n=(0,\ldots,0,1)\in\mathbb{S}^{n-1}$. Generalized Khavinson conjecture. For every $q\in[1,\infty]$, (i) for every $x\in\mathbb{B}^{n}\backslash\{0\}$,

$$\mathbf{C}_{\mathbb{B}^{n},q}(x)=\mathbf{C}_{\mathbb{B}^{n},q}\left(x;\pm\frac{x}{|x|}\right);$$

and (ii) for every $x\in\mathbb{H}^{n}$,

$$\mathbf{C}_{\mathbb{H}^{n},q}(x)=\mathbf{C}_{\mathbb{H}^{n},q}(x;\pm e_n).$$

The conjecture extends Khavinson's sharp radial-derivative estimate for bounded harmonic functions in $\mathbb{B}^{3}$ to a statement that the extremal direction for the full gradient is radial in the ball and vertical in the half-space. The source notes that it dates back to 1992 and attributes the original conjectural gradient statement to Khavinson in private conversations with Gresin and Maz'ya; no resolution is supplied here.