The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

Technical Summary: The Doubling Property of the Elliptic Measure with Averaged Diverging Drifts

Problem Statement
This paper investigates the doubling property of the elliptic measure associated with a linear elliptic operator $L$ in divergence form with a drift term, defined on a 1-sided chord arc domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$):
$$Lu = -\nabla \cdot (A\nabla u) + B \cdot \nabla u$$
The central challenge addressed is the behavior of the elliptic measure when the drift term $B$ satisfies a "smallness" condition in an averaged sense over Whitney cubes, rather than the stricter pointwise smallness condition previously assumed in the literature. Specifically, the paper considers drifts that may diverge like the inverse distance to the boundary ($\delta(x)^{-1}$) but satisfy an average bound:
$$\int_{U_Q} |B|^2 \delta(X) \, dV \leq \beta^2 l(Q)^{n-1}$$
where $U_Q$ is a fattened Whitney cube associated with a boundary cube $Q$, and $\beta$ is a small constant. This generalizes earlier results (e.g., Hofmann-Lewis [HL01]) which required the pointwise estimate $|B(x)| \leq \beta/\delta(x)$. The paper also addresses the existence of solutions for the adjoint operator $L^*$, which is necessary for establishing lower bounds on the Green's function.

Methodology
The proof strategy relies on establishing two main ingredients: pointwise estimates for the Green's function and a Bourgain-type estimate for the elliptic measure.

1. Existence and Regularity: The author first outlines the existence of weak solutions for both $L$ and its adjoint $L^*$ using the Lax-Milgram lemma. A crucial step involves proving the coercivity of the associated bilinear forms. For the adjoint operator, an additional condition on the divergence of the drift, $-\frac{M}{\delta(x)^2} \leq (\nabla \cdot B) \leq 0$, is required to ensure coercivity.
2. Hardy Inequalities: The paper proves a local and global Hardy inequality in 1-sided chord arc domains with Ahlfors-David regular (ADR) boundaries. Unlike previous derivations that relied on global versions, this work provides a direct proof using a stopping time argument on dyadic cubes, which is noted as a novel contribution to the literature.
3. Boundary Hölder Regularity: Adapting techniques from Hofmann-Lewis [HL01], the author establishes boundary Hölder continuity for solutions vanishing on a surface ball. This requires handling the drift term $B$ via a decomposition of Whitney cubes. The proof splits cubes into two categories based on the support of a test function and utilizes the averaged smallness condition (Eq. 2) alongside the pointwise bound (Eq. 4) to control the term $\int |B|^2 u^2 \psi^2$.
4. Green's Function Estimates:
   • Lower Bounds: Following the approach of Grüter and Widman [GW82], the paper derives pointwise lower bounds for the Green's function $G(x,y)$ assuming the existence of the adjoint Green's function.
   • Upper Bounds: Deriving upper bounds is more delicate due to the lack of pointwise smallness. The author employs a Calderón-Zygmund decomposition of the drift term $|B|^2$ within a Whitney cube. This allows the isolation of a "bad set" of cubes where the drift might be large, while ensuring the drift remains pointwise small on the "good set." By constructing a test function supported on the good set and utilizing the global Hardy inequality, the author establishes the necessary upper bounds.
5. Bourgain Estimate and Doubling: Combining the boundary Hölder regularity with the Green's function estimates, the paper derives a Bourgain-type estimate (a uniform lower bound on the elliptic measure of surface balls). Finally, following the argument in Aikawa [Ai06], these ingredients are combined to prove the doubling property of the elliptic measure.

Key Contributions and Results

• Generalization of Drift Conditions: The primary result (Theorem 3) establishes the doubling property of the elliptic measure under the assumption that the drift satisfies an average smallness condition (Eq. 2) on Whitney cubes, rather than the pointwise condition (Eq. 3). This is a significant relaxation of hypotheses, even for the half-space.
• New Hardy Inequality Proof: The paper provides a direct proof of local and global Hardy inequalities in 1-sided chord arc domains using a stopping time argument, distinct from previous indirect proofs.
• Green's Function Bounds: The paper establishes both pointwise upper and lower bounds for the Green's function under the averaged drift condition. The upper bound proof notably utilizes a Calderón-Zygmund decomposition to manage the lack of pointwise control on the drift.
• Adjoint Existence: The work clarifies the conditions required for the existence of solutions to the adjoint equation, specifically highlighting the necessity of the divergence condition on $B$ for the adjoint's bilinear form.

Significance and Claims
The paper claims that these results are new even for the half-space, extending the perturbative theory of rough Dirichlet solvability. The work connects to recent developments in solvability for operators with Carleson measure conditions on lower-order terms (e.g., Hofmann-Lewis, Kenig-Pipher, and recent work by the author and others).

The author positions this work as a foundational step toward extending the results of [HP24] (which established $L^p$ Dirichlet solvability) to operators where the supremum condition within the Carleson measure integral is removed. The paper explicitly states that while the doubling property is established here, the full adaptation of methods to prove Dirichlet solvability under this more general Carleson measure condition on the drift will be the subject of future work.

The significance lies in the ability to handle drifts that are not pointwise small but are "small on average," thereby broadening the class of elliptic operators for which the doubling property of the elliptic measure—and consequently, the solvability of boundary value problems—can be guaranteed.