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

TL;DR

This paper refines spectral multiplier theorems for Grushin operators in the plane, establishing sharper conditions and precise eigenfunction estimates, leading to improved boundedness results for associated operators.

Contribution

It replaces the previous $L^ ext{infty}$ Sobolev condition with an $L^2$ condition, sharpening the spectral multiplier theorem for Grushin operators.

Findings

Sharp $L^1$ boundedness range for Bochner--Riesz means

Refined spectral multiplier conditions using $L^2$ Sobolev spaces

Precise eigenfunction estimates in transition regions

Abstract

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 $L^\infty$ 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.