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We prove a multiplier theorem of Mihlin-H\"ormander type for operators of theform $-\Delta_x - V(x) \Delta_y$ 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 $V_j$ areperturbations 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 mainnovelty of the result resides in its robustness: this appears to be the firstsharp multiplier theorem for nonelliptic subelliptic operators allowing forstep higher than two and perturbation of the coefficients. The proof hinges onprecise estimates for eigenvalues and eigenfunctions of one-dimensionalSchr\"odinger operators, which are stable under perturbations of the potential.

A robust approach to sharp multiplier theorems for Grushin operators