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

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This study shows Green functions for degenerate elliptic operators on uniformly rectifiable sets are strongly approximated by a distance function. This advances understanding of Green function behavior and set properties.

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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.