Some questions of harmonic analysis in weighted Morrey type spaces

Abstract

Weighted Morrey-type classes of functions that are harmonic in the unit disk and in the upper half plane are defined in this work. Under some conditions on the weight function, we study some properties of functions belonging to these classes. Estimation of the maximum values of harmonic functions for a nontangential angle through the Hardy-Littlewood maximal function are generalized to more general case, and then the boundedness of Hardy-Littlewood operator is applied in the Morrey-type spaces. Weighted Morrey-Lebesgue type space is defined, where the shift operator is continuous with respect to shift, and its invariance with regard to the singular operator is proved. The validity of Minkowski inequality in Morrey-Lebesgue type spaces is also proved. An approximation properties of the Poisson kernel are studied.