An Extremal Problem for Integrals on a Measure Space with Abstract Parameters

Abstract

An extremal problem for integrals on a measure space with parameters $y, y_{_0}\in Y$ is considered, where Y is a set of points. A class of integrands for which the integral attains its supremum over $y\in Y$ at a fixed point $y_{_0}$ is described. The integrands of such kind in ${{\mathbb {R}}}^n$ and on the unit sphere ${\mathbb S}^{n-1}$ in ${{\mathbb {R}}}^n$ with vector parameters are pointed out. As applications, we give a new simple proof of the sharp real-part estimate for analytic functions from the Hardy spaces in the upper half-plane as well as a solution of the extremal problem for some integrals on ${{\mathbb {S}}}^{n-1}$ with vector parameters. As consequence, we find the sharp constant in a pointwise estimate for solutions of the Lamé system in the upper half-space of ${\mathbb R}^n$ with boundary data from $L^p$ for the case $p=(n+2m)/(2m)$, where m is a positive integer.