A Green function characterization of uniformly rectifiable sets of any codimension
Joseph Feneuil, Linhan Li
TL;DR
This paper provides a unified Green function-based characterization of uniformly rectifiable sets across all codimensions, extending previous results to higher codimension boundaries and more general elliptic operators.
Contribution
It introduces a novel Green function criterion for uniform rectifiability applicable to any codimension, generalizing prior work and offering new insights for higher codimension boundaries.
Findings
Unified characterization for all codimensions
Extension of Azzam's result to general elliptic operators
New results for boundaries with codimension greater than 1
Abstract
In this paper, we obtain a unified characterization of uniformly rectifiable sets of {\it any codimension} in terms of a Carleson estimate on the second derivatives of the Green function. When restricted to domains with boundaries of codimension 1, our result generalizes a previous result of Azzam for the Laplacian to more general elliptic operators. For domains with boundaries of codimension greater than 1, our result is completely new.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Nonlinear Partial Differential Equations · Analytic and geometric function theory