Lebesgue inequalities for poisson integrals

Abstract

We obtain estimates for the deviations of the Fourier partial sums on the sets of the Poisson integrals of functions from the spaceL _p ,p≥1, that are expressed in terms of the values of the best approximations of such functions by trigonometric polynomials in the metric ofL _p . We show that the estimates obtained are unimprovable on some important functional subsets.