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 $$\partial_tu- \Delta u=0, \quad \text{in} R^{n+1}\setminus E$$ satisfies Carleson measure estimate condition. An important technical novelty of our work is we develop corona domain approximation scheme for $E$ in terms regular Lip(1/2,1) graph domains. This has an analogous elliptic version which improvement the known results setting.