Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces

TL;DR

This paper thoroughly investigates the optimal boundedness conditions of weighted Hardy operators on rearrangement-invariant spaces, aiming to identify the best possible spaces and norms for practical applications in analysis.

Contribution

It characterizes the optimal rearrangement-invariant spaces and norms ensuring boundedness of weighted Hardy operators, including their iterated forms, and explores their simplification for practical use.

Findings

Identified optimal rearrangement-invariant spaces for Hardy operators.

Derived simplified expressions for complex optimal norms.

Extended analysis to iterated weighted Hardy-type operators.

Abstract

The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied with emphasis being put on the optimality of the obtained results. First, the optimal rearrangement-invariant function spaces (that is, the best possible function spaces within the class of rearrangement-invariant function spaces) guaranteeing the boundedness of the operators from/to a given rearrangement-invariant function space are described. Second, the optimal rearrangement-invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions, arguably more useful in practice, is addressed. Last, iterated weighted Hardy-type operators are also studied. Besides aiming to provide a comprehensive treatment of the optimal behavior of the operators on rearrangement-invariant function spaces in one place, the paper is motivated by its applicability in various fields of mathematical analysis, such as harmonic analysis, extrapolation theory or the theory of Sobolev-type spaces.