Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces

TL;DR

This paper establishes the strong parabolic Harnack inequality for heat equations on fractal-like metric measure spaces satisfying certain geometric and functional inequalities, extending classical results to more general, non-geodesic settings.

Contribution

It proves the parabolic Harnack inequality for nonsymmetric heat equations on fractal-type spaces with minimal geometric assumptions, including volume doubling and Poincaré inequalities.

Findings

Proves strong parabolic Harnack inequality for local weak solutions.

Derives weighted Poincaré inequality and heat kernel bounds.

Extends classical results to non-geodesic, fractal-like metric measure spaces.

Abstract

This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincar\'e inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincar\'e inequality, as well as upper and lower bounds for non-symmetric heat kernels.