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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.19393 |
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| _version_ | 1866914777556582400 |
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| author | Chen, Hua Chen, Hong-Ge Li, Jin-Ning |
| author_facet | Chen, Hua Chen, Hong-Ge Li, Jin-Ning |
| contents | Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of Hörmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(Ω)$, where $Ω\subset\subset U$ is a general open bounded subset of $U$. By employing Rothschild-Stein's lifting technique and saturation method, we prove the representation formula for smooth functions with compact support in $Ω$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(Ω)$, where the critical Sobolev exponent depends on the generalized Métivier index. As applications of these sharp Sobolev inequalities, we establish the isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and Moser-Trudinger inequality in the context of general Hörmander vector fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp embedding results and geometric inequalities for Hörmander vector fields Chen, Hua Chen, Hong-Ge Li, Jin-Ning Analysis of PDEs 35J70, 35H20, 46E35 Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of Hörmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(Ω)$, where $Ω\subset\subset U$ is a general open bounded subset of $U$. By employing Rothschild-Stein's lifting technique and saturation method, we prove the representation formula for smooth functions with compact support in $Ω$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(Ω)$, where the critical Sobolev exponent depends on the generalized Métivier index. As applications of these sharp Sobolev inequalities, we establish the isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and Moser-Trudinger inequality in the context of general Hörmander vector fields. |
| title | Sharp embedding results and geometric inequalities for Hörmander vector fields |
| topic | Analysis of PDEs 35J70, 35H20, 46E35 |
| url | https://arxiv.org/abs/2404.19393 |