Weighted Hardy Inequalities for Nested Averages

Abstract: We study a family of Hardy-type inequalities for weighted averages over nested subsets of a measure space. Given a partition of a measure space and a weight function $m$, we consider operators of the form [ f \mapsto \frac{1}{M_n}\int_{X^{(n)}} m(x)f(x)\,\mathrm{d}μ(x), ] with additional weights on the resulting sequence of averages. In particular, we generalize an inequality obtained by Vincent and Sohani in \cite{VincentSohani2025} and characterize the boundedness in terms of the finiteness of a single testing quantity $β$. We also provide two-sided estimates for the best constant $C_{\mathrm{opt}}$, namely [ β\leq C_{\mathrm{opt}} \leq p^{1/q} (p')^{1/p'}β\leq 2β. ] Thus the characterization is never off by more than a factor of 2. We also develop a second approach, inspired by Broadbent's proof of Hardy's inequality, which gives a local sufficient condition that often provides sharper constants and recovers several important cases, including the classical weighted Hardy inequality.