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The Condition, -Approximators, and Varopoulos Extensions in Uniform Domains.

S Bortz1, B Poggi2, O Tapiola2

  • 1Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 USA.

Journal of Geometric Analysis
|May 13, 2024
PubMed
Summary
This summary is machine-generated.

We establish a quantitative link between elliptic measures and surface measures in uniform domains. This finding enables boundary data to have smooth extensions, even on unrectifiable boundaries.

Keywords:
Carleson measureElliptic measureTheVaropoulos extension

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Area of Science:

  • Real Analysis
  • Partial Differential Equations
  • Harmonic Analysis

Background:

  • Uniform domains with n-Ahlfors regular boundaries are crucial in analysis.
  • Elliptic operators and their associated elliptic measures are fundamental objects of study.
  • Carleson measures play a key role in understanding function spaces and boundary behavior.

Purpose of the Study:

  • To establish a quantitative equivalence between the absolute continuity of elliptic measure with respect to surface measure and the approximability of solutions to elliptic equations.
  • To investigate the existence of Varopoulos-type extensions for boundary data with compact support, even in domains with unrectifiable boundaries.

Main Methods:

  • Utilizing the concept of $\epsilon$-approximability for bounded solutions of elliptic equations.
  • Characterizing $\epsilon$-approximability through the lens of Carleson measures with controlled norms.
  • Extending recent results on Varopoulos-type extensions to a broader class of domains.

Main Results:

  • A uniform domain with an n-Ahlfors regular boundary has an elliptic measure quantitatively absolutely continuous with respect to its surface measure if and only if bounded solutions to the associated elliptic equation are $\epsilon$-approximable.
  • $\epsilon$-approximability is defined via the existence of a function whose difference from the solution defines a Carleson measure with controlled norms.
  • Boundary functions with compact support admit Varopoulos-type extensions on sets with potentially unrectifiable boundaries, satisfying controlled Carleson measure estimates.

Conclusions:

  • The study provides a precise quantitative relationship between elliptic measures and surface measures.
  • The findings extend the theory of boundary behavior of solutions to elliptic equations to more general domains.
  • This work deepens the understanding of function extensions and their properties in harmonic analysis.