Аңдатпа
The well-known Hardy inequalities, formulated in both continuous and discrete cases, play an important role in mathematical analysis, differential equations and many other branches of mathematics. The original forms of these inequalities were subsequently extended and generalized in various directions, leading to the development of Hardy inequalities as an independent and significant area of research. A central problem in the theory of weighted inequalities is the characterization of conditions under which inequalities involving Hardy-type operators hold. Many cases of weighted estimates for linear integral Hardy-type operators have been considered, and there is a large number of books and scientific articles on this topic. More recently, considerable attention has been given to iterated Hardy-type operators due to their application in Morrey-type spaces. This paper analyzes a class of operators formed by iterating two operators, one of which involves a kernel satisfying conditions that generalize those considered previously. The study examines Hardy-type inequalities associated with these iterated operators and establishes necessary and sufficient conditions for their validity. The characterization of weighted Hardy inequalities involving iterated operators can now be applied to study of bilinear weighted Hardy-type inequalities.