Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

TL;DR

This paper establishes Carleson measure estimates for caloric functions in domains with parabolic uniformly rectifiable boundaries, introducing a novel corona domain approximation scheme for such sets.

Contribution

It develops a new corona domain approximation scheme for parabolic uniformly rectifiable sets, extending elliptic techniques to the parabolic setting.

Findings

Proves Carleson measure estimates for bounded caloric functions.

Introduces a corona domain approximation scheme for parabolic UR sets.

Provides an elliptic analogue improving previous results.

Abstract

Let $E \subset \mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$\partial_tu- \Delta u=0, \quad \text{in} \quad \mathbb R^{n+1}\setminus E$$ satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for $E$ in terms of regular Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic version which is an improvement of the known results in that setting.