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

Abstract

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on 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 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 halfspace.

Sharp estimates for the gradient of solutions to the heat equation

“…In the present paper we find the best coefficients in certain inequalities for solutions to the heat equation. Previously results of similar nature for stationary problems were obtained in our works [1]- [4] and [6], where solutions of the Laplace, Lamé and Stokes equations were considered.…”