Widder's representation theorem for symmetric local Dirichlet spaces

TL;DR

This paper extends Widder's representation theorem, originally for the heat equation in Euclidean space, to symmetric local Dirichlet spaces satisfying a local parabolic Harnack inequality, broadening its applicability.

Contribution

It establishes an analogous Widder's theorem for local weak solutions of the heat equation on symmetric local Dirichlet spaces, under specific regularity conditions.

Findings

Representation theorem holds in symmetric local Dirichlet spaces

Applicable to solutions satisfying local parabolic Harnack inequality

Generalizes classical PDE results to abstract spaces

Abstract

In classical PDE theory, Widder's theorem gives a representation for nonnegative solutions of the heat equation on $\mathbb{R}^n$. We show that an analogous theorem holds for local weak solutions of the canonical "heat equation" on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.