About the convergence to initial data of the heat problem on the Heisenberg group

TL;DR

This paper establishes integrability conditions for initial data to ensure solutions to the heat equation on the Heisenberg group, characterizes weighted Lebesgue spaces for solution existence, and proves boundedness of associated maximal functions.

Contribution

It provides new integrability criteria and characterizations for solutions of the heat problem on the Heisenberg group, including weighted space conditions and maximal function bounds.

Findings

Identified integrability conditions for initial data

Characterized weighted Lebesgue spaces for solution existence

Proved boundedness of local maximal functions with weights

Abstract

We find integrability conditions on the initial data $f$ for the existence of solutions of the Heat problem on the Heisenberg group. From this result we characterize the weighted Lebesgue spaces for which the solutions exists a.e. when the time goes to zero. Finally we also obtain boundedness of the local maximal function associated to the heat kernel with weights.