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An analyst’s traveling salesman theorem for sets of dimension larger than one

- Jonas Azzam, R. Schul
- Mathematics
- 9 September 2016

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… Expand

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

- Jonas Azzam, R. Schul
- Mathematics
- 21 May 2011

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… Expand

Characterization of n-rectifiability in terms of Jones’ square function: Part II

- Jonas Azzam, X. Tolsa
- Mathematics
- 7 January 2015

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… Expand

Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability

- Jonas Azzam, Mihalis Mourgoglou, X. Tolsa
- Mathematics
- 3 February 2016

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… Expand

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

- Jonas Azzam, M. Hall, R. Strichartz
- Mathematics
- 28 November 2007

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… Expand

Bounded mean oscillation and the uniqueness of active scalar equations

- Jonas Azzam, J. Bedrossian
- Mathematics
- 12 August 2011

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… Expand

Absolute continuity of harmonic measure for domains with lower regular boundaries

- M. Akman, Jonas Azzam, Mihalis Mourgoglou
- MathematicsAdvances in Mathematics
- 24 May 2016

On a two-phase problem for harmonic measure in general domains

- Jonas Azzam, Mihalis Mourgoglou, X. Tolsa, A. Volberg
- MathematicsAmerican Journal of Mathematics
- 20 September 2016

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… Expand

Semi-Uniform Domains and the A∞ Property for Harmonic Measure

- Jonas Azzam
- MathematicsInternational 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… Expand

Sets of Absolute Continuity for Harmonic Measure in NTA Domains

- Jonas Azzam
- Mathematics
- 10 October 2014

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… Expand

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