2018/08/27 by Guy David, David, Guy, Max Engelstein +3
Computer Science · Mathematics · #35R35 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1808.08882
openalex publication_date 2018/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we characterize the rectifiability (both uniform and not) of\nan Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a\nregularized distance function in the complement of that set. In particular, we\nestablish a certain version of the Riesz transform characterization of\nrectifiability for lower-dimensional sets. We also uncover a special situation\nin which the regularized distance is itself a solution to a degenerate elliptic\noperator in the complement of E. This allows us to precisely compute the\nharmonic measure of those sets associated to this degenerate operator and prove\nthat, in a sharp contrast with the usual setting of co-dimension one, a\nconverse to the Dahlberg's theorem (see [Da] and [DFM2]) must be false on lower\ndimensional boundaries without additional assumptions.\n