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

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.