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An analyst’s traveling salesman theorem for sets of dimension larger than one
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $$\beta $$β-numbers. These $$\beta $$β-numbers are geometric quantities
Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be
Characterization of n-rectifiability in terms of Jones’ square function: Part II
We show that a Radon measure $${\mu}$$μ in $${\mathbb{R}^d}$$Rd which is absolutely continuous with respect to the n-dimensional Hausdorff measure $${\mathcal{H}^n}$$Hn is n-rectifiable if the so
Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability
We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with
Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket
On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy e ϕ and conformal Laplacian Δ ϕ for a given conformal factor ϕ, based on the corresponding notions in
Bounded mean oscillation and the uniqueness of active scalar equations
We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we
On a two-phase problem for harmonic measure in general domains
abstract:We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and
Semi-Uniform Domains and the A∞ Property for Harmonic Measure
  • Jonas Azzam
  • Mathematics
    International Mathematics Research Notices
  • 8 November 2017
We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [ 5] that, for John domains satisfying the capacity density condition (CDC), the doubling property
Sets of Absolute Continuity for Harmonic Measure in NTA Domains
We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then ω|E≪ℋd|E$\omega |_{E}\ll \mathcal {H}^{d}|_{E}$. Moreover, this holds quantitatively