Convolution inequalities in weighted Lorentz spaces: case 0<q<1

Abstract

Operators acting on function spaces are classical subjects of study in functional analysis. This thesis contributes to the research on this topic, focusing particularly on integral and supremal operators and weighted function spaces.Proving boundedness conditions of a convolution-type operator between weighted Lorentz spaces is the first type of a problem investigated here. The results have a form of weighted Young-type convolution inequalities, addressing also optimality properties of involved domain spaces. In addition to that, the outcome includes an overview of basic properties of some new function spaces appearing in the proven inequalities. Product-based bilinear and multilinear Hardy-type operators are another matter of focus. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable. The last part of the presented work concerns iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, completing the theory of the involved fundamental Hardy-type operators.