H\"older regularity for parabolic De Giorgi classes in metric measure spaces

TL;DR

This paper proves that functions in parabolic De Giorgi classes are H"older continuous within metric measure spaces that satisfy certain measure and inequality conditions, extending regularity results to more general spaces.

Contribution

It establishes H"older regularity for parabolic De Giorgi classes in metric measure spaces under minimal geometric and measure assumptions.

Findings

Proves H"older continuity of functions in the classes

Extends regularity theory to metric measure spaces

Uses doubling measure and Poincaré inequality assumptions

Abstract

We give a proof for the H\"older continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak $(1,p)$-Poincar\'e inequality and to satisfy the annular decay property.