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.

arXiv comment39 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 (11)

• Analysis of joint spectral multipliers on Lie groups of polynomial growth
• Spectral analysis and the Aharonov-Bohm~effect on certain almost-Riemannian manifolds
• Spectral multipliers on $2$-step groups: topological versus homogeneous dimension
• A sharp multiplier theorem for Grushin operators in arbitrary dimensions
• Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres
• Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres
• On Geometric Quantum Confinement in Grushin-type Manifolds
• From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere
• Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces
• Spectral multipliers for the Kohn Laplacian on forms on the sphere in ${\mathbb{C}}^n$
• A robust approach to sharp multiplier theorems for Grushin operators

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