Overview
The research investigates the doubling property of the elliptic measure associated with specific elliptic operators. These operators possess an elliptic principal term and a drift component. A key characteristic of this drift is its average divergence on Whitney cubes, where the divergence rate is proportional to the inverse distance to the boundary, accompanied by a small constant.
Research Context
The study builds upon and extends earlier work concerning elliptic measures and operators. Previous findings in this area often relied on a hypothesis of pointwise smallness for the drift. This current research refines that condition, suggesting that a small Carleson constant assumption on the drift can achieve similar results. This generalization connects to recent perturbative findings related to rough Dirichlet solvability in domains where drifts or potentials adhere to a Carleson measure condition. The concept of Carleson measure conditions in this context has been explored previously by Hofmann-Lewis and Kenig-Pipher.
The investigation specifically operates within 1-sided chord arc domains. The reported results are presented as novel, even when applied to the simpler case of a half-space.
Approach
The core approach involves analyzing an operator defined by an elliptic principal term and a drift. The drift's behavior is characterized by an average divergence on Whitney cubes, specifically diverging like the inverse distance to the boundary, scaled by a small constant. This condition on the drift is described as essentially a small Carleson constant assumption.
As part of the methodology, the researchers also established Hardy inequalities within the specified 1-sided chord arc domains, which feature Alhfors-David regular boundaries. The derivation of these Hardy inequalities employed a stopping time argument.
The research's contribution lies in demonstrating how the doubling property holds under this generalized drift condition, moving from a pointwise smallness requirement to an average divergence criterion linked to a Carleson constant.
Findings
- The elliptic measure demonstrates the doubling property for operators with an elliptic principal term and a drift.
- The drift in question diverges on average on Whitney cubes.
- The average divergence rate of the drift is proportional to the inverse distance to the boundary, qualified by a small constant.
- This finding is supported by a small Carleson constant assumption on the drift.
- The results generalize previous findings that required a hypothesis of pointwise smallness for the drift.
- New Hardy inequalities were proven in 1-sided chord arc domains with Alhfors-David regular boundaries.
- These Hardy inequalities were established using a stopping time argument.
- The results are considered new, even for the half-space.
Why This Matters
This research refines the conditions under which the doubling property of the elliptic measure holds, transitioning from a stringent pointwise smallness requirement to a more general average divergence criterion for the drift. This generalization contributes to an expanded understanding of rough Dirichlet solvability in domains with specific drift or potential characteristics, referencing prior work by Hofmann-Lewis and Kenig-Pipher.