Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

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.