A sharp multiplier theorem for Grushin operators in arbitrary dimensions

Abstract

In a recent work by A. Martini and A. Sikora, sharp $L^p$ spectral multiplier theorems for the Grushin operators acting on $\mathbb R^{d_1}_{x'} \times \mathbb R^{d_2}_{x''}$ and defined by the formula

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 essentially on the second factor of $\mathbb R^{d_1} \times \mathbb R^{d_2}$ (in contrast to the mentioned work, where the "extra weights" depend only on the first factor) and gives a new unified proof of the sharp results without restrictions on the dimensions.