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 sole assumptions that $V(-x) \simeq xV'(x)$ 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 independent $V$. The assumption second derivative $V''$ can actually weakened to H\"older-type condition $V'$. proof hinges analysis one-dimensional Schr\"odinger operators, including universal estimates eigenvalue gaps matrix coefficients potential.