Weighted Hardy Inequalities for Nested Averages

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 $$f\mapsto \frac{1}{M_n}{\int }_{X^{(n)}}m(x)f(x)\mathrm{d}\mu (x),$$ 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 $\beta$. We also provide two-sided estimates for the best constant $C_{\mathrm{o}\mathrm{p}\mathrm{t}}$, namely $$\beta \le C_{\mathrm{o}\mathrm{p}\mathrm{t}}\le p^{1\mathrm{/}q}(p^{\mathrm{\prime }}{)}^{1\mathrm{/}{p}^{\mathrm{\prime }}}\beta \le 2\beta \mathrm{.}$$ 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.