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

An extremal problem for integrals on a measure space with parameters y,y0∈Y is considered, where Y is a set of points. A class of integrands for which the integral attains its supremum over y∈ Y at a fixed point y0 is described. The integrands of such kind in Rⁿ and on the unit sphere S^{n - 1} in Rⁿ 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 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 Rⁿ with boundary data from L^p for the case p= (n+ 2 m) / (2 m) , where m is a positive integer.