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 \colon \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–Hörmander 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 Hölder-type condition on V' . The proof hinges on the spectral analysis of one-dimensional Schrödinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potential.