  We find optimal integrability conditions on the initial data $f$ for the
existence of solutions $e^{-t\Delta_{\lambda}}f(x)$ and
$e^{-t\sqrt{\Delta_{\lambda}}}f(x)$ of the heat and Poisson initial data
problems for the Bessel operator $\Delta_{\lambda}$ in $\mathbb{R}^{+}$. We
also characterize the most general class of weights $v$ for which the solutions
converge a.e. to $f$ for every $f\in L^{p}(v)$, with $1\le p<\infty$. Finally,
we show that for such weights and $1<p<\infty$ the local maximal operators are
bounded from $L^{p}(v)$ to $L^{p}(u)$, for some weight $u$.
