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 ${\mathbb{R}}_{x^{\prime }}^{d_1}\times {\mathbb{R}}_{x^{\prime \prime }}^{d_2}$ and defined by the formula

are obtained in the case $d_1\ge d_2$. 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.