[Paper Review] Dahlberg's theorem in higher co-dimension

Guy David, Joseph Feneuil+1 more|arXiv (Cornell University)|Apr 3, 2017

Advanced Harmonic Analysis Research31 references3 citations

TL;DR

This paper establishes the first analogue of Dahlberg's theorem in higher co-dimension by constructing a linear degenerate elliptic operator $ L $ on the complement of a $ d $-dimensional Lipschitz graph in $ \mathbb{R}^n $, $ d < n-1 $, with small Lipschitz constant, such that the associated harmonic measure $ \omega_L $ is absolutely continuous with respect to the $ d $-dimensional Hausdorff measure. The key result is a sufficient condition on the coefficient matrix of $ L $ ensuring mutual absolute continuity of $ \omega_L $ and $ \mathcal{H}^d|_\Gamma $, extending classical boundary value theory beyond co-dimension one.

ABSTRACT

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n-1$ in $\\mathbb R^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond. In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph $\\Gamma$ of dimension $d$ in $\\mathbb R^n$, $d&lt;n-1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator $L$ such that the corresponding harmonic measure $\\omega_L$ is absolutely continuous with respect to the Hausdorff measure on $\\Gamma$. More generally, we provide sufficient conditions on the matrix of coefficients of $L$ which guarantee the mutual absolute continuity of $\\omega_L$ and the Hausdorff measure.

Motivation & Objective

• To extend Dahlberg's classical theorem on absolute continuity of harmonic measure with respect to surface measure from co-dimension one to higher co-dimensions.
• To establish the existence of a degenerate elliptic operator $ L $ on $ \mathbb{R}^n \setminus \Gamma $, where $ \Gamma $ is a $ d $-dimensional Lipschitz graph with $ d < n-1 $, such that the harmonic measure $ \omega_L $ is absolutely continuous with respect to $ \mathcal{H}^d|_\Gamma $.
• To provide sufficient conditions on the coefficient matrix of $ L $ ensuring mutual absolute continuity of $ \omega_L $ and $ \mathcal{H}^d|_\Gamma $, generalizing the classical theory to higher co-dimension settings.
• To develop a new framework involving change of variables, soft distance functions, and square function estimates to control oscillations and prove the $ A^\infty $ property of harmonic measure in this non-classical setting.

Proposed method

• Constructing a bi-Lipschitz change of variables $ \rho $ that maps the $ d $-dimensional graph $ \Gamma $ to a flat $ \mathbb{R}^d $, enabling the transfer of geometric and analytic properties.
• Defining a soft distance function $ D $ to control the geometry of the domain and ensure uniform regularity of the Jacobian under the change of variables.
• Using $ \alpha $-numbers and Wasserstein distances to quantify the regularity of the soft distance and control the Carleson measure condition for the Jacobian of the transformation.
• Conjugating the original elliptic operator $ L $ via the change of variables to obtain a new operator $ L_0 $, whose coefficients satisfy controlled degeneracy and allow for $ L^2 $ square function estimates.
• Applying Harnack chain arguments and pointwise estimates on harmonic functions to control oscillations of solutions in dyadic Whitney-type regions.
• Establishing the $ A^\infty $ property of harmonic measure by proving that the square function $ S^{Q_r}(x) $ satisfies a Carleson measure estimate, which implies the desired absolute continuity.

Experimental results

Research questions

• RQ1 Can Dahlberg's theorem on the absolute continuity of harmonic measure with respect to surface measure be extended to domains whose boundaries have co-dimension greater than one?
• RQ2 What conditions on the coefficient matrix of a degenerate elliptic operator $ L $ ensure that the associated harmonic measure $ \omega_L $ is mutually absolutely continuous with respect to the $ d $-dimensional Hausdorff measure on a $ d $-dimensional Lipschitz graph $ \Gamma \subset \mathbb{R}^n $, $ d < n-1 $?
• RQ3 How can one construct a degenerate elliptic operator on $ \mathbb{R}^n \setminus \Gamma $ such that the harmonic measure is absolutely continuous with respect to $ \mathcal{H}^d|_\Gamma $, even when the boundary is not of co-dimension one?
• RQ4 What geometric and analytic tools are necessary to control oscillations of harmonic functions and prove the $ A^\infty $ property in higher co-dimension?
• RQ5 Is it possible to transfer the classical theory of harmonic measure on NTA domains to higher co-dimension settings using a change of variables and degenerate elliptic operators?

Key findings

• The paper constructs a linear degenerate elliptic operator $ L $ on $ \mathbb{R}^n \setminus \Gamma $, where $ \Gamma $ is a $ d $-dimensional Lipschitz graph with $ d < n-1 $ and small Lipschitz constant, such that the harmonic measure $ \omega_L $ is absolutely continuous with respect to the $ d $-dimensional Hausdorff measure $ \mathcal{H}^d|_\Gamma $.
• A sufficient condition on the coefficient matrix of $ L $ is provided, ensuring mutual absolute continuity of $ \omega_L $ and $ \mathcal{H}^d|_\Gamma $, generalizing the classical result to higher co-dimensions.
• The authors prove that the $ A^\infty $ property of harmonic measure holds for such operators, by showing that the square function $ S^{Q_r}(x) $ satisfies a Carleson measure estimate with a constant depending only on the ellipticity constants and dimension.
• The key estimate $ \frac{|E|}{|\Delta_r|} \leq C \frac{\log \varepsilon_0}{\log \delta} $ is established, where $ E \subset \Delta_r $ satisfies $ \omega(E) \leq \delta \omega(\Delta_r) $, and this implies $ |E| < \varepsilon |\Delta_r| $ for sufficiently small $ \delta $, proving the $ A^\infty $ property.
• The proof relies on a novel change of variables $ \rho $, a soft distance function $ D $, and a dyadic covering argument with controlled overlap, enabling the control of oscillations and the derivation of square function estimates.
• The result is robust under small perturbations of the graph and holds uniformly for all $ d < n-1 $, with all constants depending only on the ellipticity constants, dimension $ n $, and the co-dimension $ n-d $.