Carleman estimates for higher step Grushin operators
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913231255109632 |
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| author | De Bie, Hendrik Lian, Pan |
| author_facet | De Bie, Hendrik Lian, Pan |
| contents | The higher step Grushin operators $Δ_α$ are a family of sub-elliptic operators which degenerate on a sub-manifold of $\mathbb{R}^{n+m}$. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted $L^{p}-L^{q}$ estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators $-Δ_α+V$ at points of the degeneracy manifold, where $V$ belongs to certain $ L^{r}_{\rm loc}(\mathbb{R}^{n+m})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Carleman estimates for higher step Grushin operators De Bie, Hendrik Lian, Pan Analysis of PDEs Classical Analysis and ODEs Primary 35H20, 35B60, Secondary 33C45 The higher step Grushin operators $Δ_α$ are a family of sub-elliptic operators which degenerate on a sub-manifold of $\mathbb{R}^{n+m}$. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted $L^{p}-L^{q}$ estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators $-Δ_α+V$ at points of the degeneracy manifold, where $V$ belongs to certain $ L^{r}_{\rm loc}(\mathbb{R}^{n+m})$. |
| title | Carleman estimates for higher step Grushin operators |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary 35H20, 35B60, Secondary 33C45 |
| url | https://arxiv.org/abs/2402.07348 |