Hardy inequalities in normal form

TL;DR

This paper introduces a unified normal form for Hardy operators, simplifying the theory of weighted Hardy inequalities across various spaces and establishing new estimates for the best constants involved.

Contribution

It presents a general transition to normal form applicable to diverse Hardy operators, preserving key properties and extending known results with new estimates.

Findings

Unified normal form for Hardy operators across multiple spaces

Preservation of boundedness, compactness, and operator norm during transition

New precise estimates for the best constants in Hardy inequalities

Abstract

A simple normal form for Hardy operators is introduced that unifies and simplifies the theory of weighted Hardy inequalities. A straightforward transition to normal form is given that applies to the various Hardy operators and their duals, whether defined on Lebesgue spaces of sequences, of functions on the half-line, or of functions on $\mathbb R^n$ or more general metric spaces. This is done by introducing an abstract formulation of Hardy operators, more general than any of these, and showing that the normal form transition applies to all operators formulated in this way. The transition to normal form is shown to preserve boundedness, compactness, and operator norm. To a large extent the transition can be carried out via well-behaved linear operators. Known results for boundedness and compactness of Hardy operators are given simple proofs and extended, via the transition, to this general setting. New estimates for the best constant in Hardy inequalities are established and a large class of Hardy inequalities is identified in which the best constants are is known precisely.