5 problems

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

Potential-kernel conjecture. For $x\in B^+$ and $y\in S^+$,

Poisson kernel expansion conjecture. The Poisson kernel of any Riemannian manifold has the form displayed above for suitable $U_j$.

Final conjecture. The estimates from the heat-kernel theorem and the Poisson-kernel theorem hold for all $\nu>-1$.

Let $\gamma$ be a non-negative integer, and let $F_\gamma$ be the Poisson kernel under consideration. For $1\leq\beta\leq\gamma+2$, let $f_\beta(x)$ be polynomials. Explicit…