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 $$|u(x)|\le \frac{C_p}{(1-|x{|}^2{)}^{(n-1)/p}}\Vert u{\Vert }_{h^p(B^n)},u\in h^p(B^n),x\in B^n,$$ in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler, Bourdon and Ramey (\cite{ABR}), where they obtained similar results which are sharp only in the cases $p=2$ and $p=1$.

9 pages

References in corpus (1)

• Optimal estimates for the gradient of harmonic functions in the unit disk