On singularity of $p$-energy measures on metric measure spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916742520897536 |
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| author | Yang, Meng |
| author_facet | Yang, Meng |
| contents | For $p\in(1,+\infty)$, we prove that for a $p$-energy on a metric measure space, under the volume doubling condition, the conjunction of the Poincaré inequality and the cutoff Sobolev inequality both with $p$-walk dimension strictly larger than $p$ implies the singularity of the associated $p$-energy measure with respect to the underlying measure. We also prove that under the slow volume regular condition, the conjunction of the Poincaré inequality and the cutoff Sobolev inequality is equivalent to the resistance estimate. As a direct corollary, on a large family of fractals and metric measure spaces, including the Sierpiński gasket and the Sierpiński carpet, we obtain the singularity of the $p$-energy measure with respect to the underlying measure for all $p$ strictly great than the Ahlfors regular conformal dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On singularity of $p$-energy measures on metric measure spaces Yang, Meng Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 For $p\in(1,+\infty)$, we prove that for a $p$-energy on a metric measure space, under the volume doubling condition, the conjunction of the Poincaré inequality and the cutoff Sobolev inequality both with $p$-walk dimension strictly larger than $p$ implies the singularity of the associated $p$-energy measure with respect to the underlying measure. We also prove that under the slow volume regular condition, the conjunction of the Poincaré inequality and the cutoff Sobolev inequality is equivalent to the resistance estimate. As a direct corollary, on a large family of fractals and metric measure spaces, including the Sierpiński gasket and the Sierpiński carpet, we obtain the singularity of the $p$-energy measure with respect to the underlying measure for all $p$ strictly great than the Ahlfors regular conformal dimension. |
| title | On singularity of $p$-energy measures on metric measure spaces |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 |
| url | https://arxiv.org/abs/2505.12468 |