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 Lame 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.