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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.09503 |
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| _version_ | 1866912924607447040 |
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| author | Yang, Meng |
| author_facet | Yang, Meng |
| contents | For $p>1$, we introduce the cutoff Sobolev inequality on general metric measure spaces, and prove that there exists a metric measure space endowed with a $p$-energy that satisfies the chain condition, the volume regular condition with respect to a doubling scaling function $Φ$, and that both the Poincaré inequality and the the cutoff Sobolev inequality with respect to a doubling scaling function $Ψ$ hold if and only if $$\frac{1}{C}\left(\frac{R}{r}\right)^p\le\frac{Ψ(R)}{Ψ(r)}\le C\left(\frac{R}{r}\right)^{p-1}\frac{Φ(R)}{Φ(r)}\text{ for any }r\le R.$$ In particular, given any pair of doubling functions $Φ$ and $Ψ$ satisfying the above inequality, we construct a metric measure space endowed with a $p$-energy on which all the above conditions are satisfied. As a direct corollary, we prove that there exists a metric measure space which is $d_h$-Ahlfors regular and has $p$-walk dimension $β_p$ if and only if $$p\leβ_p\le d_h+(p-1).$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan [Ann. Probab., to appear]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09503 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-Poincaré inequalities and cutoff Sobolev inequalities on metric measure spaces Yang, Meng Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 For $p>1$, we introduce the cutoff Sobolev inequality on general metric measure spaces, and prove that there exists a metric measure space endowed with a $p$-energy that satisfies the chain condition, the volume regular condition with respect to a doubling scaling function $Φ$, and that both the Poincaré inequality and the the cutoff Sobolev inequality with respect to a doubling scaling function $Ψ$ hold if and only if $$\frac{1}{C}\left(\frac{R}{r}\right)^p\le\frac{Ψ(R)}{Ψ(r)}\le C\left(\frac{R}{r}\right)^{p-1}\frac{Φ(R)}{Φ(r)}\text{ for any }r\le R.$$ In particular, given any pair of doubling functions $Φ$ and $Ψ$ satisfying the above inequality, we construct a metric measure space endowed with a $p$-energy on which all the above conditions are satisfied. As a direct corollary, we prove that there exists a metric measure space which is $d_h$-Ahlfors regular and has $p$-walk dimension $β_p$ if and only if $$p\leβ_p\le d_h+(p-1).$$ Our proof builds on the Laakso-type space theory, which was recently developed by Murugan [Ann. Probab., to appear]. |
| title | $p$-Poincaré inequalities and cutoff Sobolev inequalities on metric measure spaces |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 31E05, 28A80 |
| url | https://arxiv.org/abs/2504.09503 |