Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators
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| Format: | Preprint |
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2020
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| _version_ | 1866917687013146624 |
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| author | Casarino, Valentina Ciatti, Paolo Martini, Alessio |
| author_facet | Casarino, Valentina Ciatti, Paolo Martini, Alessio |
| contents | We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For these operators we prove a spectral multiplier theorem of Mihlin-Hörmander type, which is optimal whenever $2k \leq d$, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_02916 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators Casarino, Valentina Ciatti, Paolo Martini, Alessio Analysis of PDEs Classical Analysis and ODEs 33C55, 42B15, 43A85 (Primary) 53C17, 58J50 (Secondary) We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For these operators we prove a spectral multiplier theorem of Mihlin-Hörmander type, which is optimal whenever $2k \leq d$, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials. |
| title | Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators |
| topic | Analysis of PDEs Classical Analysis and ODEs 33C55, 42B15, 43A85 (Primary) 53C17, 58J50 (Secondary) |
| url | https://arxiv.org/abs/2009.02916 |