A sharp multiplier theorem for a perturbation-invariant class of Grushin operators of arbitrary step

Gian Maria Dall'Ara, Alessio Martini

Published 2017-12-08Version 1

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 $d_1 \geq (1+\sigma) d_2$. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schr\"odinger operators, which are stable under perturbations of the potential.