Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918297258164224 |
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| author | Bui, The Anh Duong, Xuan Thinh Merz, Konstantin |
| author_facet | Bui, The Anh Duong, Xuan Thinh Merz, Konstantin |
| contents | We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of $L^p$-Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e., ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift Bui, The Anh Duong, Xuan Thinh Merz, Konstantin Analysis of PDEs Mathematical Physics Functional Analysis We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of $L^p$-Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e., ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup. |
| title | Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2410.00191 |