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

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