WEIGHTED ESTIMATE OF A MATRIX OPERATOR WITH VARIABLE UPPER LIMIT ON THE CONE OF MONOTONE SEQUENCES

Hardy's inequality was formulated in 1920 and finally proved in 1925. Since then, this inequality has been significantly developed. The first development was related to the consideration of more general weights. The next step was to use more general operators with other kernels instead of the Hardy operator. Currently, there are many works devoted to Hardy-type inequalities with iterated operators. Motivated by important applications, all these generalizations of Hardy's inequality are studied not only on the cone of non-negative functions, but also on the cone of monotone functions. In this paper, we consider the problem of finding necessary and sufficient conditions for the fulfillment of a weighted Hardy-type inequality on the cone of monotone sequences for 1<p≤q<∞. The main method for solving the problem is the reduction method, which, using the Sawyer principle, allows us to reduce a Hardy-type inequality on the cone of monotone sequences to some inequality for all non-negative sequences.