Optimal behavior of weighted Hardy operators on rearrangement‐invariant spaces

Abstract

The behavior of certain weighted Hardy‐type operators on rearrangement‐invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal 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 is addressed. Next, the relation between optimal rearrangement‐invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy‐type operators are also studied.