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

TL;DR

This paper establishes optimal conditions on initial data for the convergence of heat and Poisson solutions related to the Bessel operator, including weight characterizations and boundedness of maximal operators.

Contribution

It provides the first comprehensive characterization of integrability and weight conditions ensuring pointwise convergence for Bessel operator heat and Poisson problems.

Findings

Optimal integrability conditions for initial data.

Characterization of weights for almost everywhere convergence.

Boundedness of local maximal operators between weighted spaces.

Abstract

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 $u$.