Two-Weight Hardy Inequality on Topological Measure Spaces

We consider a Hardy type integral operator \ (T \) associated with a family of open subsets \ ( (t) \) of an open set \ (\) in a Hausdorff topological space \ (X \). In the inequality\ (_ |Tf (x) |q u (x) \, d (x) ) ^1/q C (_ |f (x) |ᵖ v (x) \, d (x) ) ^1/p, measures \ (, \) are \ (\) -additive Borel measures; the weights \ (u, v \) are positive and finite almost everywhere, \ (1 0 \) is independent of \ (f, u, v, , \). 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.