Hardy Averaging via Distribution Transport: Sharp Lp Bounds, Power Weights, and a Volterra Resolvent

Hardy-type averaging operators arise in real analysis, rearrangement theory, weighted inequalities, and Volterra integral equations. This paper develops a distribution-function transport on (0,∞) equipped with an atomless Borel measure μ, showing that the cumulative map Φ(x)=μ((0,x]) implements a measure isomorphism onto Lebesgue measure under transparent support and continuity hypotheses. Under this transport, the Hardy averaging operator relative to μ is conjugate to the classical Hardy operator on (0,∞) with Lebesgue measure. The main contribution is the systematic transport principle: classical constants, extremizing sequences, weighted criteria, endpoint estimates, and resolvent information are transferred exactly to the μ-scale. We establish sharp Lp(μ) bounds, sharp power-weight extensions in Lp(Φγdμ) for −1<γ<p−1, a transported one-weight Hardy class beyond powers, endpoint weak and strong estimates, spectral interpretation of the Volterra threshold, and numerical illustrations for the transported constants and a Volterra feedback equation.