Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

TL;DR

This paper derives explicit sharp coefficient estimates for generalized Poisson integrals in half-space, providing new insights into harmonic and biharmonic functions with applications to classical conjectures.

Contribution

It introduces explicit formulas for sharp coefficients in generalized Poisson integrals, extending estimates to harmonic and biharmonic functions in the half-space.

Findings

Explicit formulas for sharp coefficients at p=1, p=2, and some p=∞ cases.

Conditions for analogues of Khavinson's conjecture in the generalized setting.

Applications to sharp estimates for harmonic and biharmonic functions.

Abstract

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function $f$ on ${\mathbb R}^{n-1}$ is obtained under the assumption that $f$ belongs to $L^p$. It is assumed that the kernel of the integral depends on the parameters $\alpha$ and $\beta$. The explicit formulas for the sharp coefficients are found for the cases $p=1$, $p=2$ and for some values of $\alpha , \beta$ in the case $p=\infty$. Conditions ensuring the validity of some analogues of the Khavinson's conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.