Two-weight Hardy inequality on topological measure spaces

Abstract

We consider a Hardy type integral operator \( T \) associated with a family of open subsets \( \Omega(t) \) of an open set \( \Omega \) in a Hausdorff topological space \( X \). In the inequality

\[\left( \int_\Omega |Tf(x)|^q u(x)\,d\mu(x) \right)^{1/q} \leq C \left( \int_\Omega |f(x)|^p v(x)\,d\nu(x) \right)^{1/p},\]

the measures \( \mu, \nu \) are \( \sigma \)-additive Borel measures; the weights \( u, v \) are positive and finite almost everywhere, \( 1 < p < \infty \), \( 0 < q < \infty \), and \( C > 0 \) is independent of \( f, u, v, \mu, \nu \). We find necessary and sufficient conditions for the boundedness and compactness of the operator \( T \) and obtain two-sided estimates for its approximation numbers. All results are proved using domain partitions, thus providing a roadmap for generalizing many one-dimensional results to a Hausdorff topological space.