From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

Abstract

We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in ${\mathbb{R}}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.

32 pages

References in corpus (4)

• 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
• Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces
• Lp estimates for the wave equation associated to the Grushin operator

Cited by in corpus (9)

• Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
• An optimal multiplier theorem for Grushin operators in the plane, I
• On $L^2$-boundedness of pseudo-multipliers associated to the Grushin operator
• A robust approach to sharp multiplier theorems for Grushin operators
• Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators
• Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups
• Uniform pointwise estimates for ultraspherical polynomials
• Null controllability of the parabolic spherical Grushin equation
• Bilinear Bochner-Riesz Means on Métivier groups