GENERALIZED POISSON INTEGRAL AND SHARP ESTIMATES FOR HARMONIC AND BIHARMONIC FUNCTIONS IN THE HALF-SPACE

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on f(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 = infinity. Conditions ensuring the validity of some analogues of the Khavinsons conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.
Funding Agencies|Ministry of Education and Science of the Russian Federation [02.a03.21.0008]