  We establish that the spectral multiplier $\frak{M}(G_{\alpha})$ associated
to the differential operator $$ G_{\alpha}=- \Delta_x
+\sum_{j=1}^m{{\alpha_j^2-1/4}\over{x_j^2}}-|x|^2 \Delta_y \; \text{on}
(0,\infty)^m \times \R^n,$$ which we denominate Bessel-Grushin operator, is of
weak type $(1,1)$ provided that $\frak{M}$ is in a suitable local Sobolev
space. In order to do this we prove a suitable weighted Plancherel estimate.
Also, we study $L^p$-boundedness properties of Riesz transforms associated to
$G_{\alpha}$, in the case $n=1$.
