Three weight Hardy inequality on measure topological spaces

For the Hardy inequality to hold on a Hausdorff topological space, we obtain necessary and sufficient conditions on the weights and measures. As in the recent paper by G. Sinnamon (2022), we assume total orderedness of the family of sets that generate the Hardy operator. Sinnamon’s method consists in the reduction of the problem to an equivalent one-dimensional problem. We provide a different, direct proof which develops the approach suggested by D. Prokhorov (2006) in the one-dimensional case.