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On the Mattila-Sjolin theorem for distance sets

- A. Iosevich, Mihalis Mourgoglou, K. Taylor
- Mathematics
- 31 October 2011

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set… Expand

Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems

- P. Auscher, Mihalis Mourgoglou
- Mathematics
- 18 June 2013

The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We… Expand

Improved Cotlar's inequality in the context of local $Tb$ theorems

- Henri Martikainen, Mihalis Mourgoglou, X. Tolsa
- Mathematics
- 9 December 2015

We prove in the context of local $Tb$ theorems with $L^p$ type testing conditions an improved version of Cotlar's inequality. This is related to the problem of removing the so called buffer… Expand

Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

- J. Garnett, Mihalis Mourgoglou, X. Tolsa
- MathematicsDuke Mathematical Journal
- 1 November 2016

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded… 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

Representation and uniqueness for boundary value elliptic problems via first order systems

- P. Auscher, Mihalis Mourgoglou
- MathematicsRevista Matemática Iberoamericana
- 10 April 2014

Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary… Expand

Absolute continuity of harmonic measure for domains with lower regular boundaries

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

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors… Expand

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

On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators

- A. Iosevich, Mihalis Mourgoglou, E. Palsson
- Mathematics
- 31 October 2011

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive… Expand

Rectifiability of harmonic measure in domains with porous boundaries

- Jonas Azzam, Mihalis Mourgoglou, X. Tolsa
- Mathematics
- 22 May 2015

We show that if $n\geq 1$, $\Omega\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite and positive Hausdorff $H^{n}$-measure upon… Expand

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