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–Hörmander type, which is optimal whenever 2k≤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.
Original languageEnglish
Article numberrnab007
JournalInternational Mathematics Research Notices
Early online date10 Mar 2021
DOIs
Publication statusE-pub ahead of print - 10 Mar 2021

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