Solvability of the Lp Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

• 成果类型: Article
• 署名作者: Bortz, Simon; Hofmann, Steve; Martell, Jose Maria; Nystrom, Kaj
• 署名单位: University of Alabama System; University of Alabama Tuscaloosa; University of Missouri System; University of Missouri Columbia; Universidad Carlos III de Madrid; Complutense University of Madrid; Consejo Superior de Investigaciones Cientificas (CSIC); CSIC - Instituto de Ciencia de Materiales de Madrid (ICMM); CSIC - Instituto de Ciencias Matematicas (ICMAT); Autonomous University of Madrid; Uppsala University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01300-1
• 发表日期: 2025
• 页码: 165-217
• 关键词: SINGULAR-INTEGRALS; HARMONIC-MEASURES; BOUNDARY; REGULARITY; EXISTENCE; BEHAVIOR

摘要

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is a parabolic A(infinity) weight with respect to surface measure (which in turn is equivalent to L(p )solvability of the Dirichlet problem for some finite p), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the A(infinity) property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has L(p )solvability (for some p < infinity) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type.