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

¹ Department of Mathematics, Ariel University, Ariel 40700, Israel.
² Department of Mathematical Sciences, University of Liverpool, M & O Building, Liverpool L69 3BX, UK
³ Department of Mathematics, Linköping University, 58183 Linköping, Sweden
⁴ RUDN University, 6 Miklukho-Maklay St., Moscow 117198, Russia

^* Corresponding author: kresin@ariel.ac.il

Received: 25 November 2017
Accepted: 25 November 2017

Abstract

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on ℝⁿ⁻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 α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. 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.