Mihalis Mourgoglou

activity

20102026

most citedRectifiability of harmonic measure

61 citations · 237 across the 33 of their papers we have counts for

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33 papers · 1 filter

math.CA2025

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^{p}$ Hajlasz-Sobolev space on the boundary from the space of functions o…

math.CA2023

Quantitative differentiability on uniformly rectifiable sets

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

We prove $L^{p}$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-typ…

math.CA2019★ 58 cited

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ -solvability of the Dirichlet problem

Jonas Azzam, Steve Hofmann, José María Martell +2

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure, on the b…

math.CA2018

Failure of $L^{2}$ boundedness of gradients of single layer potentials for measures with zero low density

José M. Conde-Alonso, Mihalis Mourgoglou, Xavier Tolsa

Consider a totally irregular measure $μ$ in $R^{n + 1}$ , that is, the upper density $lim sup_{r \to 0} \frac{μ ( B ( x , r ))}{( 2 r ) ^{n}}$ is positive $μ$ -a.e.\ in $R^{n + 1}$ , an…

math.CA2016★ 29 cited

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

John Garnett, Mihalis Mourgoglou, Xavier Tolsa

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

math.CA2016★ 18 cited

Uniform rectifiability, elliptic measure, square functions, and $ε$ -approximability via an ACF monotonicity formula

Jonas Azzam, John Garnett, Mihalis Mourgoglou +1

Let $Ω \subset R^{n + 1}$ , $n \geq 2$ , be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ i…

math.CA2016

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

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa +1

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 rec…

math.CA2016★ 2 cited

Non-homogeneous square functions on general sets: suppression and big pieces methods

Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $T b$ theorems. The setting is new: we consider conical square f…

math.CA2016★ 3 cited

The one-phase problem for harmonic measure in two-sided NTA domains

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

We show that if $Ω \subset R^{3}$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $VMO (σ)$ , where $σ$ is th…

math.CA2016

Absolute continuity of harmonic measure for domains with lower regular boundaries

Murat Akman, Jonas Azzam, Mihalis Mourgoglou

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

math.CA2016

Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

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 absolut…

math.CA2016

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

Mihalis Mourgoglou

Let $Ω \subset R^{n + 1}$ , $n \geq 1$ , be an open and connected set. Set $T_{n}$ to be the set of points $ξ \in \partial Ω$ so that there exists an approximate tangent…

math.CA2015

Improved Cotlar's inequality in the context of local $T b$ theorems

Henri Martikainen, Mihalis Mourgoglou, Xavier Tolsa

We prove in the context of local $T b$ theorems with $L^{p}$ type testing conditions an improved version of Cotlar's inequality. This is related to the problem of removing the so call…

math.CA2015

A new approach to non-homogeneous local $T b$ theorems

Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen

We develop a new general method to prove various non-doubling local Tb theorems. The method combines the non-homogeneous good lambda method of Tolsa, the big pieces Tb theorem of N…

math.CA2015★ 2 cited

Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure

Jonas Azzam, Steve Hofmann, José María Martell +4

In the present paper we sketch the proof of the fact that for any open connected set $Ω \subset R^{n + 1}$ , $n \geq 1$ , and any $E \subset \partial Ω$ with $0<\mathcal{H}^n(E)<\…

math.CA2015

Harmonic measure and Riesz transform in uniform and general domains

Mihalis Mourgoglou, Xavier Tolsa

Let $Ω ⊊ R^{n + 1}$ be open and let $μ$ be some measure supported on $\partial Ω$ such that $μ (B (x, r)) \leq C r^{n}$ for all $x \in R^{n + 1}$ , $r > 0$ . We show that if…

math.CA2015★ 61 cited

Rectifiability of harmonic measure

Jonas Azzam, Steve Hofmann, José María Martell +4

In the present paper we prove that for any open connected set $Ω \subset R^{n + 1}$ , $n \geq 1$ , and any $E \subset \partial Ω$ with $H^{n} (E) < \infty$ , absolute continui…

math.CA2015

Tangent measures and absolute continuity of harmonic measure

Jonas Azzam, Mihalis Mourgoglou

We show that for uniform domains $Ω \subseteq R^{d + 1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely contin…

math.CA2015★ 7 cited

Uniform domains with rectifiable boundaries and harmonic measure

Mihalis Mourgoglou

We assume that $Ω \subset R^{d + 1}$ , $d \geq 2$ , is a uniform domain with lower $d$ -Ahlfors-David regular and $d$ -rectifiable boundary. We show that if $\mathcal{H}^d|_{\par…

math.CA2015★ 6 cited

Rectifiability of harmonic measure in domains with porous boundaries

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

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

math.CA2015★ 1 cited

Note about square function estimates and uniformly rectifiable measures

Henri Martikainen, Mihalis Mourgoglou

We generalise and offer a different proof of a recent $L^{2}$ square function estimate on UR sets by Hofmann, Mitrea, Mitrea and Morris. The proof is a short argument using the $α$ -n…

math.CA2015

Singular sets for harmonic measure on locally flat domains with locally finite surface measure

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that…

math.CA2014

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

Pascal Auscher, Mihalis Mourgoglou

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 classe…

math.CA2014

Boundedness of non-homogeneous square functions and $L^{q}$ type testing conditions with $q \in (1, 2)$

Henri Martikainen, Mihalis Mourgoglou

We continue the study of local $T b$ theorems for square functions defined in the upper half-space $(R_{+}^{n + 1}, μ \times d t / t)$ . Here $μ$ is allowed to be a non-homogeneous…

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Quantitative differentiability on uniformly rectifiable sets

Harmonic measure and quantitative connectivity: geometric characterization of the Lp-solvability of the Dirichlet problem

Failure of L2 boundedness of gradients of single layer potentials for measures with zero low density

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

Uniform rectifiability, elliptic measure, square functions, and ε-approximability via an ACF monotonicity formula

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

Non-homogeneous square functions on general sets: suppression and big pieces methods

The one-phase problem for harmonic measure in two-sided NTA domains

Absolute continuity of harmonic measure for domains with lower regular boundaries

Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

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

A new approach to non-homogeneous local Tb theorems

Harmonic measure is rectifiable if it is absolutely continuous with respect to the co-dimension one Hausdorff measure

Harmonic measure and Riesz transform in uniform and general domains

Rectifiability of harmonic measure

Tangent measures and absolute continuity of harmonic measure

Uniform domains with rectifiable boundaries and harmonic measure

Rectifiability of harmonic measure in domains with porous boundaries

Note about square function estimates and uniformly rectifiable measures

Singular sets for harmonic measure on locally flat domains with locally finite surface measure

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

Boundedness of non-homogeneous square functions and Lq type testing conditions with q∈(1,2)

Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$ -solvability of the Dirichlet problem

Failure of $L^{2}$ boundedness of gradients of single layer potentials for measures with zero low density

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

Uniform rectifiability, elliptic measure, square functions, and $ε$ -approximability via an ACF monotonicity formula

Improved Cotlar's inequality in the context of local $T b$ theorems

A new approach to non-homogeneous local $T b$ theorems

Boundedness of non-homogeneous square functions and $L^{q}$ type testing conditions with $q \in (1, 2)$