Corona Decompositions for Parabolic Uniformly Rectifiable Sets

Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

“…To give an idea of the methods involved in the proof of Theorem 1.1, the primary novelty of our work is a corona domain approximation scheme (Proposition 3.25) in terms of regular Lip( 1 / 2 , 1) graph domains. This is in contrast to the (elliptic) NTA domain approximations produced in [Hofmann et al 2016] for uniformly rectifiable sets. In fact, our proof here carries over without modification to the elliptic setting, 3 providing an (improved) approximation by Lipschitz domains.…”

“…To deal with this difficulty, we are forced to build appropriate approximating domains with better properties than would be enjoyed by the parabolic analogues of the chord-arc domains constructed in that paper. In particular, our construction improves on that of [Hofmann et al 2016], even in the elliptic setting. We shall discuss these issues in more detail momentarily.…”

“…In this area of research, the lack of smoothness presents itself in the structure or regularity of the coefficients of the operator, or in the geometry of the domain. Recently, sustained efforts in this area have provided characterizations of quantitative geometric notions (e.g., uniform rectifiability) in terms of quantitative estimates for harmonic functions [Garnett et al 2018;Hofmann et al 2016] and a geometric characterization of the L p -solvability of the Dirichlet problem [Azzam et al 2020]. This paper concerns the parabolic analogue of [Hofmann et al 2016] and overcomes the substantial difficulty introduced by the distinguished time direction and the anisotropic scaling.…”

“…Recently, sustained efforts in this area have provided characterizations of quantitative geometric notions (e.g., uniform rectifiability) in terms of quantitative estimates for harmonic functions [Garnett et al 2018;Hofmann et al 2016] and a geometric characterization of the L p -solvability of the Dirichlet problem [Azzam et al 2020]. This paper concerns the parabolic analogue of [Hofmann et al 2016] and overcomes the substantial difficulty introduced by the distinguished time direction and the anisotropic scaling. To deal with this difficulty, we are forced to build appropriate approximating domains with better properties than would be enjoyed by the parabolic analogues of the chord-arc domains constructed in that paper.…”

“…In fact, our proof here carries over without modification to the elliptic setting, 3 providing an (improved) approximation by Lipschitz domains. In [Hofmann et al 2016] the authors use Whitney cubes to construct these NTA domains using dyadic sawtooths and exploiting an elliptic bilateral corona decomposition. The heuristic in the elliptic setting is that these sawtooth domains inherit many essential properties of the original boundary.…”