  In a recent work, P. Chen and E. M. Ouhabaz proved a $p$-specific
$L^p$-spectral multiplier theorem for the Grushin operator acting on
$\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$ which is given by \[ L
=-\sum_{j=1}^{d_1} \partial_{x_j}^2 - \bigg( \sum_{j=1}^{d_1} |x_j|^2\bigg)
\sum_{k=1}^{d_2}\partial_{y_k}^2. \] Their approach yields an $L^p$-spectral
multiplier theorem within the range $1< p\le \min\{
\frac{2d_1}{d_1+2},\frac{2(d_2+1)}{d_2+3} \}$ under a regularity condition on
the multiplier which is sharp only when $d_1\ge d_2$. In this paper, we improve
on this result by proving $L^p$-boundedness under the expected sharp regularity
condition $s>(d_1+d_2)(1/p-1/2)$. Our approach avoids the usage of weighted
restriction type estimates which played a key role in the work of P. Chen and
E. M. Ouhabaz, and is rather based on a careful analysis of the underlying
sub-Riemannian geometry and restriction type estimates where the multiplier is
truncated along the spectrum.
