A sharp multiplier theorem for Grushin operators in arbitrary dimensions

TL;DR

This paper extends sharp spectral multiplier theorems for Grushin operators to the case where the first dimension is less than the second, using novel weighted estimates for a unified proof.

Contribution

It completes the spectral multiplier results for Grushin operators by addressing the case where $d_1 < d_2$, providing a new proof method with weighted estimates.

Findings

Sharp spectral multiplier theorems established for all dimension cases

New weighted estimate approach depending on the second factor

Unified proof technique without dimension restrictions

Abstract

In a recent work by A. Martini and A. Sikora (arXiv:1204.1159), sharp L^p spectral multiplier theorems for the Grushin operators acting on $R^{d_1} \times R^{d_2}$ are obtained in the case $d_1 \geq d_2$. Here we complete the picture by proving sharp results in the case $d_1 < d_2$. Our approach exploits L^2 weighted estimates with "extra weights" depending only on the second factor of $R^{d_1} \times R^{d_2}$ (in contrast with the mentioned work, where the "extra weights" depend on the first factor) and gives a new unified proof of the sharp results without restrictions on the dimensions.