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

In a previous work we proved a spectral multiplier theorem of Mihlin–H"ormander type for two-dimensional Grushin operators $-\partial_x^2 - V(x) \partial_y^2$,where$V$ is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of $V$. Here we refine this result, by replacing the $L^{\mathrm{\infty }}$ Sobolev condition on the multiplier with a sharper $L^2$ condition. As a consequence, we obtain the sharp range of $L^1$ boundedness for the associated Bochner–Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schr"odinger operators with doubling single-well potentials.