A robust approach to sharp multiplier theorems for Grushin operators

We prove a multiplier theorem of Mihlin-H"ormander type for operators of the form $-{\mathrm{\Delta }}_x-V(x){\mathrm{\Delta }}_y$ 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 $t\mapsto \mathrm{\mid }t{\mathrm{\mid }}^{2\sigma }$, and $\sigma \in (1/2,\infty)$\mathrm{.}Theresultissharpwhenever$d_1 \geq \sigma d_2$. 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.