Energy inequalities for cutoff functions of $p$-energies on metric measures spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914170597801984 |
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| author | Yang, Meng |
| author_facet | Yang, Meng |
| contents | For $p>1$, and for a $p$-energy on a metric measure space, we provide various geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy inequalities for cutoff functions of $p$-energies on metric measures spaces Yang, Meng Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 For $p>1$, and for a $p$-energy on a metric measure space, we provide various geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608]. |
| title | Energy inequalities for cutoff functions of $p$-energies on metric measures spaces |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 |
| url | https://arxiv.org/abs/2507.08577 |