Green function estimates on complements of low-dimensional uniformly rectifiable sets.

Area of Science:

• Harmonic Analysis
• Partial Differential Equations
• Geometric Measure Theory

Background:

• Recent work established weak approximation of Green functions on uniformly rectifiable sets.
• Uniform rectifiability of a set is linked to Green function estimates.
• Degenerate elliptic operators and domains with lower-dimensional boundaries are key areas of study.

Purpose of the Study:

• To investigate a strong analogue of existing weak Green function approximation results.
• To analyze Green function behavior for degenerate elliptic operators on domains with uniformly rectifiable boundaries.
• To establish a Carleson measure estimate for the Green function in this context.

Main Methods:

• Utilizing intricate integration by parts techniques.
• Leveraging the properties of the distance function to the boundary.
• Analyzing degenerate elliptic operators associated with uniformly rectifiable sets.

Main Results:

• The Green function G for the operator L, with a pole at infinity, is well approximated by multiples of the boundary distance function.
• The function $\frac{G(x)}{ \mapsto(x)}$ satisfies a Carleson measure estimate on the boundary.
• This provides a strong analogue to previously established weak approximation results.

Conclusions:

• The findings establish a strong approximation for Green functions in this setting.
• The methods differ significantly from prior work, relying on integration by parts rather than compactness arguments.
• The results deepen the connection between Green function properties and the geometry of the domain boundary.