  In this paper we give a criterion to prove boundedness results for several
operators from $H^1((0,\infty),\gamma_\alpha)$ to
$L^1((0,\infty),\gamma_\alpha)$ and also from
$L^\infty((0,\infty),\gamma_\alpha)$ to $\BMO((0,\infty),\gamma_\alpha)$, with
respect to the probability measure $d\gamma_\alpha
(x)=\frac{2}{\Gamma(\alpha+1)} x^{2\alpha+1} e^{-x^2} dx$ on $(0,\infty)$ when
${\alpha>-\frac12}$. We shall apply it to establish endpoint estimates for
Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of
Laplace transform type, fractional integrals and variation operators in the
Laguerre setting.
