Analysis of and on uniformly rectifiable sets

  title={Analysis of and on uniformly rectifiable sets},
  author={Guy David and S. Semmes},
The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant substitute for the classical notion of rectifiability; as the answer (sometimes only conjecturally) to certain geometric questions in complex and harmonic analysis; as a condition which ensures the parametrizability of a given set, with estimates, but with some holes and self-intersections allowed; and… 
Coronizations and big pieces in metric spaces
We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the
Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do
Square functions, non-tangential limits and harmonic measure in co-dimensions larger than one.
In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement
Generalized rectifiability of measures and the identification problem
One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises
A sharp necessary condition for rectifiable curves in metric spaces
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a
Square functions, nontangential limits, and harmonic measure in codimension larger than 1
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
Sufficient conditions for C^1,α parametrization and rectifiability
  • Silvia Ghinassi
  • Mathematics
    Annales Academiae Scientiarum Fennicae Mathematica
  • 2020
We say a measure is C d-rectifiable if there is a countable union of C d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R to be C d-rectifiable,
Singular integrals and rectifiability
The problems addressed in this dissertation live in the intersection between Harmonic Analysis and Geometric Measure Theory, and so one should say that they belong to the area of Geometric Analysis.
Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress
  • S. Hofmann
  • Mathematics
    Acta Mathematica Sinica, English Series
  • 2019
It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary