EXACT DISCRETE REDUCTIONS FOR NESTED HARDY INEQUALITIES
Keywords:
Hardy inequality, nested averages, two-weight inequality, Banach function space, associate space, matrix operatorAbstract
Bouthat and Parisé (arXiv:2606.24044) characterized weighted Hardy inequalities for averages over nested sets and asked three questions, on function spaces beyond \(L^p\), on triangular kernel operators, and on sums of integrals over different measure spaces. We answer all three by the same reduction, which shows that the measure-theoretic inequality is equivalent, with the same optimal constant, to a discrete one. This settles the multi-space question and completes the exponent picture for nested averages, the case \(0<q<p\) being resolved for all \(1\le p<\infty\). For a Banach function space over the whole space, the right discrete object is a quotient functional on sequences, which cannot in general be recovered from the associate norms on the pieces. For direct sums of spaces over the pieces, the reduction leads exactly to these associate norms. Triangular kernels reduce, through an exact dual formula, to their column functionals. These are determined by the local norms alone when the kernel is block-factorized, and Oinarov-type criteria then hold for \(1<p\le q<\infty\) and \(1<q<p<\infty\).
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