A robust approach to sharp multiplier theorems for Grushin operators

Abstract

We prove a multiplier theorem of Mihlin-Hörmander type for operators of the form $-{\Delta }_x-V(x){\Delta }_y$ on ${\mathbb{R}}_x^{d_1}\times {\mathbb{R}}_y^{d_2}$, where $V(x)={\sum }_{j=1}^{d_1}{V}_j(x_j)$, the $V_j$ are perturbations of the power law $t\mapsto |t{|}^{2\sigma }$, and $\sigma \in (1/2,\infty )$. The result is sharp whenever $d_1\ge \sigma d_2$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential.

38 pages, accepted for publication in Transactions of the American Mathematical Society

References in corpus (2)

• Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres
• From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

Cited by in corpus (5)

• An optimal multiplier theorem for Grushin operators in the plane, II
• Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators
• Sparse bounds for pseudo-multipliers associated to Grushin operators, I
• Uniform pointwise estimates for ultraspherical polynomials
• Riesz transforms associated with the Grushin operator with drift