Calculations of the norms for monotone operators on the cones of functions with monotonicity properties

TL;DR

This paper develops a general method for calculating exact norms of monotone operators on cones of monotonic functions in ideal spaces, with applications to integral, dilation, and embedding operators.

Contribution

It introduces a unified approach to compute norms of various monotone operators in ideal spaces, extending to Hardy-type and weighted Lorentz space operators.

Findings

Calculated norms of integral operators on cones

Derived norms for dilation and embedding operators in weighted Lorentz spaces

Provided sharp estimates for Hardy-type operators on cones

Abstract

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.