A robust approach to sharp multiplier theorems for Grushin operators

TL;DR

This paper establishes a sharp Mihlin-H"ormander type multiplier theorem for a class of nonelliptic, subelliptic operators related to Grushin operators, allowing for higher step and coefficient perturbations, with robustness proven via eigenvalue estimates.

Contribution

It provides the first sharp multiplier theorem for high-step, perturbed nonelliptic subelliptic operators, extending previous results to more general settings.

Findings

The theorem is sharp when d_1 ≥ σ d_2.

Eigenvalue and eigenfunction estimates are stable under potential perturbations.

The approach is robust, accommodating higher step operators with variable coefficients.

Abstract

We prove a multiplier theorem of Mihlin-H\"ormander type for operators of the form $-\Delta_x - V(x) \Delta_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, 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 $d_1 \geq \sigma d_2$. 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 Schr\"odinger operators, which are stable under perturbations of the potential.