A sharp multiplier theorem for Grushin operators in arbitrary dimensions

In a recent work by A. Martini and A. Sikora, sharp Lp spectral multiplier theorems for the Grushin operators acting on Rd1x× Rd2x and defined by the formula are obtained in the case d1 ≥d2. Here we complete the picture by proving sharp results in the case d1 < d2. Our approach exploits L2 weighted estimates with -gextra weights-h depending essentially on the second factor of Rd1-Rd2 (in contrast to the mentioned work, where the -gextra weights'h depend only on the first factor) and gives a new unified proof of the sharp results without restrictions on the dimensions.