TY - JOUR
T1 - Non-local distance functions and geometric regularity
AU - Engelstein, Max
AU - Jeznach, Cole
AU - Mayboroda, Svitlana
N1 - Publisher Copyright:
© 2024 Elsevier Inc.
PY - 2024/5
Y1 - 2024/5
N2 - We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as part of a larger PDE theory, by David, Feneuil, and Mayboroda in [5]. The results apply to all dimensions and co-dimensions, require no underlying topological assumptions, and provide a surprisingly rich class of analytic characterizations of rectifiability.
AB - We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as part of a larger PDE theory, by David, Feneuil, and Mayboroda in [5]. The results apply to all dimensions and co-dimensions, require no underlying topological assumptions, and provide a surprisingly rich class of analytic characterizations of rectifiability.
KW - Regularized distance
KW - Square function estimates
KW - Uniform rectifiability
UR - http://www.scopus.com/inward/record.url?scp=85189967306&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85189967306&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2024.109649
DO - 10.1016/j.aim.2024.109649
M3 - Article
AN - SCOPUS:85189967306
SN - 0001-8708
VL - 445
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 109649
ER -