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

Abstract

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 $V(-x)\simeq V(x)\simeq x{V}^{\prime }(x)$ and $x^2|V^{\prime \prime }(x)|\lesssim V(x)$, we prove a spectral multiplier theorem of Mihlin–Hörmander type for $\mathcal{L}$, whose smoothness requirement is optimal and independent of $V$. The assumption on the second derivative $V^{\prime \prime }$ can actually be weakened to a Hölder-type condition on $V^{\prime }$. The proof hinges on the spectral analysis of one-dimensional Schrödinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potential.