The heat equation with singular potentials. II: Hypoelliptic case

TL;DR

This paper investigates the heat equation with highly singular potentials within the framework of hypoelliptic operators on graded Lie groups, extending previous work on classical heat equations to more general settings.

Contribution

It introduces a very weak solution concept for the heat equation with singular potentials in the hypoelliptic setting on graded Lie groups, expanding the scope of prior research.

Findings

Existence of very weak solutions for the heat equation with singular potentials.

Extension of classical results to hypoelliptic operators on graded Lie groups.

Generalization of previous work on the heat equation in Euclidean spaces.

Abstract

In this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends a previous work , where the classical heat equation on $\mathbb R^n$ was considered.