An optimal multiplier theorem for Grushin operators in the plane, I

Let $\mathcal{L} = -\partial_x^2 - V(x) \partial_y^2$ be the Grushin operator on $\mathbb{R}^2$ with coefficient $V : \mathbb{R} \to [0,\infty)$. Under the sole assumptions that $V(-x) \simeq V(x) \simeq xV'(x)$ and $x^2 |V''(x)| \lesssim V(x)$, we prove a spectral multiplier theorem of Mihlin--H\"ormander type for $\mathcal{L}$, whose smoothness requirement is optimal and independent of $V$. The assumption on the second derivative $V''$ can actually be weakened to a H\"older-type condition on $V'$. The proof hinges on the spectral analysis of one-dimensional Schr\"odinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potential.