Weight estimates for norms of operators with two variable limits of integration

Weight estimates for norms of operators with two variable limits of integration are presented. Given a number and a fixed function, a weighted Lebesgue space is defined as the set of all measurable functions with finite norms. A study conducted to prove the weight estimates considered the integral operator, where the boundary functions satisfy certain conditions. In particular, the important notion of a fairway function was introduced, which has been found to be continuous and strictly increasing. The fairway functions were used to obtain a concise criterion for the boundedness of the operator. The purpose of the study was to obtain a criterion for the boundedness of the geometric mean operator, which is closely limited to operator by a limit relation of the form. The study also determines necessary and sufficient conditions for the boundedness of the operator on the class of functions for fixed parameters and locally integrable weight functions.