Results for Multidimensional Hardy Operator Using Domain Partitions

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 \(X.\) We find necessary and sufficient conditions for the boundedness and compactness of the operator T and two-sided estimates for its approximation numbers. All results are obtained using domain partitions, thus providing a road map for generalizing many one-dimensional results to a Hausdorff topological space.