Optimal estimates for harmonic functions in the unit ball

TL;DR

This paper derives the exact sharp constants and functions for inequalities involving harmonic functions in the unit ball, extending previous results to all p-values and using special functions for precise characterization.

Contribution

It provides the first complete characterization of sharp constants and functions for harmonic function inequalities in the unit ball for all p, using hypergeometric and Euler functions.

Findings

Derived explicit sharp constants and functions for harmonic inequalities.

Extended previous results to all p-values beyond p=1 and p=2.

Utilized special functions for precise mathematical descriptions.

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, Bourdon and Ramey (\cite{ABR}), where they obtained similar results which are sharp only in the cases $p=2$ and $p=1$.