Hardy Inequalities on Metric Measure Spaces for Different Indices p and q

In this note, we present the results of Ruzhansky and Verma (Proc R Soc A 475(2223):20180310, 2018; Proc R Soc A 477:20210136, 2021) where we have characterized the weights for the Hardy inequalities to hold on general metric measure spaces which have polar decompositions for the cases $$1< p \leq q< \infty $$ and $$0 < q