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 $$ L=-\sum\_{j=1}^{d\_1}\partial\_{x'j}^2 - \Big(\sum{j=1}^{d\_1}|x'j|^2\Big) \sum{k=1}^{d\_2}\partial\_{x''\_k}^2 $$ 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 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.