Hardy inequalities on metric measure spaces, IV: The case p=1

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p=1 and 1 <= q<infinity. This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1<p <= q<infinity . The case p=1 requires a different argument and does not follow as the limit of known inequalities for p>1 . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case p=1 and 1 <= q<infinity.