A.e. convergence and 2-weight inequalities for Poisson-Laguerre semigroups

We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators L. From these, we solve two problems about the Poisson semigroup $e^{-t\sqrt{L}}$. First, we find the largest space of initial data $f$ so that $e^{-t\sqrt{L}}f(x)\to f(x)$ at a.e. $x$. Secondly, we characterize the largest class of weights $w$ which admit 2-weight inequalities of the form $\|\sup_{0<t\leq t_0}|e^{-t\sqrt{L}}f|\,\|_{L^p(v)}\lesssim \|f\|_{L^p(w)}$, for some other weight $v$.