  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.
