Maximal operators for the holomorphic Laguerre semigroup

For each $p$ in $[1,\infty)$ let $\mathbf{E}_p$ denote the closure of the region of holomorphy of the Laguerre semigroup $\{M^{\alpha}_t:t>0\}$ on $L^p$ with respect to the Laguerre measure $\mu_{\alpha}$. We prove weak type and strong type estimates for the maximal operator $f\mapsto \sup\{|M^{\alpha}_z f|:z\in \mathbf{E}_p\}$. In particular, we give a new proof for the weak type $1$ estimate for the maximal operator associated to $M^{\alpha}_t$. Our starting point is the well-known relationship between the Laguerre semigroup of half-integer parameter and the Ornstein-Uhlenbeck semigroup.