Absolute Continuity between the Surface Measure and Harmonic Measure Implies Rectifiability
نویسنده
چکیده
In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS
منابع مشابه
Uniform Rectifiability, Carleson Measure Estimates, and Approximation of Harmonic Functions
Let E ⊂ Rn+1, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := Rn+1 \ E satisfy Carleson measure estimates, and are “ε-approximable”. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute con...
متن کاملTHE WEAK-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY
Let E ⊂ Rn+1, n ≥ 2, be an Ahlfors-David regular set of dimension n. We show that the weak-A∞ property of harmonic measure, for the open set Ω := Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < ∞.
متن کاملHarmonic Measure Is Rectifiable If It Is Absolutely Continuous with Respect to the Co-dimension One Hausdorff Measure
In the present paper we sketch the proof of the fact that for any open connected set Ω ⊂ Rn+1, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable.
متن کاملRectifiability of Harmonic Measure in Domains with Porous Boundaries
We show that if n ≥ 1, Ω ⊂ R is a connected domain that is porous around a subset E ⊂ ∂Ω of finite and positive Hausdorff H-measure, and the harmonic measure ω is absolutely continuous with respect to H on E, then ω|E is concentrated on an n-rectifiable set.
متن کاملAn Identity with Applications to Harmonic Measure
In this note we use an elementary integral formula to give a new short proof of the theorem of B. E. J. Dahlberg [2] on the mutual absolute continuity of harmonic and surface measure on bounded Lipschitz domains. (See [4] for the relevant definitions.) The formula also provides a new proof of the so-called reverse Holder inequality (see [2]), and //-estimates for the Dirichlet problem (see [3])...
متن کامل