On the pointwise convergence to initial data of heat and Poisson problems for the Bessel operator

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 $u$.