A Note on Hardy Inequalities in Metric Measure Spaces for the Case p=1

In this note, we announce the two-weight Hardy inequalities on metric measure spaces possessing a polar decomposition for the case \(p=1\) and \(1 \le q <\infty \) obtained in the paper Ruzhansky et al. (Hardy inequalities on metric measure spaces, IV: The case \(p = 1\) . Forum Mathematicum, 2024, to appear. https://doi.org/10.1515/forum-2023-0319 .). We refer to Ruzhansky et al. (Hardy inequalities on metric measure spaces, IV: The case \(p = 1\) . Forum Mathematicum, 2024, to appear. https://doi.org/10.1515/forum-2023-0319 .) for the complete proofs and more details.