$L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915780479680512 |
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| author | Xu, Zhe-Feng Zhang, Ye |
| author_facet | Xu, Zhe-Feng Zhang, Ye |
| contents | Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06853 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces Xu, Zhe-Feng Zhang, Ye Functional Analysis 44A10, 28A50, 35A23 Motivated by the sharp constants in the $L^2$-Caffarelli--Kohn--Nirenberg (or $L^2$-CKN for short) inequalities on Euclidean spaces, we study, in a unified framework, a sequence of $L^2$-CKN inequalities on metric measure spaces. On a general metric measure space, this sequence implies a reverse volume comparison of Günther type. Moreover, on a subclass of spaces admitting the measure contraction property, we show that this sequence of $L^2$-CKN inequalities are valid if and only if the spaces are volume cones. We also provide a stability result for inequalities of this type on volume cones. |
| title | $L^2$-Caffarelli--Kohn--Nirenberg inequalities on metric measure spaces |
| topic | Functional Analysis 44A10, 28A50, 35A23 |
| url | https://arxiv.org/abs/2602.06853 |