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arXiv · 2009.02916

Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

Valentina Casarino·Paolo Ciatti·Alessio Martini
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Abstract

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-H\"ormander 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.

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Valentina Casarino, Paolo Ciatti, Alessio Martini. 2020-09-07. Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators. https://doi.org/10.1093/imrn%2Frnab007

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