Weighted Spectral Cluster Bounds and a Sharp Multiplier Theorem for Ultraspherical Grushin Operators

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-Hormander 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.