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

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 $2k\le d$, 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.

39 pages. This work extends the multiplier theorem proved in arXiv:1705.07068 and is based on estimates for ultraspherical polynomials in arXiv:2009.02323

References in corpus (1)

• Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

Cited by in corpus (3)

• Sparse bounds for pseudo-multipliers associated to Grushin operators, II
• On some operator-valued Fourier pseudo-multipliers associated to Grushin operators
• A sharp multiplier theorem for solvable extensions of Heisenberg and related groups