An optimal multiplier theorem for Grushin operators in the plane. I (Q6172746)

Summary: 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.