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 a $k$-dimensional sphere for some $k < d$. For these we prove spectral multiplier theorem Mihlin-H\"ormander type, is optimal whenever $2k \leq d$, and corresponding Bochner-Riesz summability result. The proof hinges suitable weighted cluster bounds, in turn depend precise estimates ultraspherical polynomials.