Hardy inequalities in normal form

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.