We characterize the rectifiability (both uniform and not) of an Ahlfors regular set E of arbitrary codimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in sharp contrast with the usual setting of codimension 1, a converse to Dahlberg’s theorem must be false on lower-dimensional boundaries without additional assumptions.
The authors want to thank Camille Labourie for pointing out Borwein’s work . They would also like to thank the referees, whose careful readings and comments greatly improved this paper.
David’s work was partially supported by Simons Collaborations in Mathematics and Physical Sciences (MPS) grant 601941, Agence Nationale de la Recherche (ANR) GEOMETRYA project grant ANR-12-BS01-0014, and European Community Marie Curie MANET project grant 607643 and GHAIA project H2020 grant 777822. Engelstein’s work was partially supported by a National Science Foundation (NSF) postdoctoral fellowship, NSF grant DMS-1703306, and by David Jerison’s grant DMS-1500771. Mayboroda’s work was partially supported by NSF INSPIRE grant DMS-1344235, NSF grant DMS-1839077, a Simons Foundation Fellowship, and Simons Collaborations in MPS grant 563916. This project is based upon work supported by NSF grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2017 spring semester.
"Square functions, nontangential limits, and harmonic measure in codimension larger than 1." Duke Math. J. 170 (3) 455 - 501, 15 February 2021. https://doi.org/10.1215/00127094-2020-0048