Weighted Estimates for Some Class of Quasilinear Operators

ABSTRACT The inclusion of kernels in Hardy‐type operators is a significant research direction. However, deriving results for general kernels remains an open problem. Consequently, current studies focus on Hardy‐type operators with kernels from specific classes satisfying particular conditions. This work investigates discrete Hardy‐type inequalities involving iterated operators and . The aim of the article is to obtain necessary and sufficient conditions for the inequality in the case of parameters and . For the case , the boundedness criterion for quasilinear operators and was established for a general class of matrices, and for the cases and , the criterion was obtained for matrices belonging to wider classes. These results allow us to study generalized bilinear operators. In particular, the results on quasilinear inequalities with matrices provide alternative proofs for estimates of bilinear inequalities. In addition, the importance of the obtained results lies in their application to the study of difference equations and the embedding theory of weighted Sobolev spaces.