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 $2k \leq d$, 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.

中文翻译：

---

超球面 Grushin 算子的加权谱簇边界和夏普乘子定理

我们在$d$-维球体上研究Grushin 类型的退化椭圆算子，它们在$k$-维球体上对于一些$k < 是奇异的。d$。对于这些算子，我们证明了 Mihlin-Hörmander 类型的谱乘法器定理，它在 $2k \leq d$ 时是最优的，以及相应的 Bochner-Riesz 可和性结果。证明取决于合适的加权谱簇边界，而这又取决于对超球多项式的精确估计。