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 $e^{-t{\mathrm{\Delta }}_{\lambda }}f(x)$ and $e^{-t\sqrt{{\mathrm{\Delta }}_{\lambda }}}f(x)$ of the heat and Poisson initial data problems for the Bessel operator ${\mathrm{\Delta }}_{\lambda }$ 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 $1\le p<\mathrm{\infty }$. Finally, we show that for such weights and $1<p<\mathrm{\infty }$ the local maximal operators are bounded from $L^p(v)$ to $L^p(u)$, for some weight $u$.