# A Green function characterization of uniformly rectifiable sets of any codimension

## 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.

## Key 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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Source: https://tomesphere.com/paper/2302.14087