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

Abstract

In a previous work we proved a spectral multiplier theorem of Mihlin--Hörmander 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 Schrödinger operators with doubling single-well potentials.

21 pages. This work refines the multiplier theorem proved in arXiv:2107.12015

References in corpus (1)

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

Cited by in corpus (5)

• An optimal multiplier theorem for Grushin operators in the plane, II
• Sparse bounds for pseudo-multipliers associated to Grushin operators, I
• Sparse bounds for pseudo-multipliers associated to Grushin operators, II
• On some operator-valued Fourier pseudo-multipliers associated to Grushin operators
• Riesz transforms associated with the Grushin operator with drift