Optimal estimates for harmonic functions in the unit ball

Abstract

We find the sharp constants C _p and the sharp functions C _p  = C _p (x) in the inequality

$$ |u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}} \|u\|_{h^{p}(B^{n})}, u\in h^{p}(B^{n}), x\in B^{n}, $$

in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler et al. (Harmonic function theory, New York, 1992), where they obtained similar results which are sharp only in the cases p = 2 and p = 1.