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In this paper, we investigate the two-weight Hardy inequalities on metricmeasure space possessing polar decompositions for the case $p=1$ and $1 \leq q<\infty.$ This result complements the Hardy inequalities obtained in \cite{RV}in the case $1< p\le q<\infty.$ The case $p=1$ requires a different argumentand does not follow as the limit of known inequalities for $p>1.$ As abyproduct, we also obtain the best constant in the established inequality. Wegive examples obtaining new weighted Hardy inequalities on homogeneous Liegroups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case$p=1$ and $1\le q<\infty.$

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