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 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} . 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 < ∞ {1<p\leq q<\infty} . The case p = 1 {p=1} requires a different argument and does not follow as the limit of known inequalities for p > 1 {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 {p=1} and 1 ≤ q < ∞ {1\leq q<\infty} .