Searcharxiv⌕ Search

arXiv subjects

Mihalis Mourgoglou

Publications and source records attributed to Mihalis Mourgoglou.

At least 37 records · Page 2Linked to original sources
arXiv2018

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

Consider a totally irregular measure $μ$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ is positive $μ$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ vanishes $μ$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_μf(x)=\int K(x,y)\,dμ(y)$ is an operator whose kernel $K(\cdot,\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then $T_μ$ is not bounded in $L^2(μ)$. This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform.

math.CA↗
arXiv2017

Tangent measures of elliptic harmonic measure and applications

Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $ Ω\subset \mathbb{R}^{n+1}$: 1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for {\it any} domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$. 2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. 3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if $ω$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.

math.AP↗
arXiv2017

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

Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ is $\varepsilon$-approximmable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called $S<N$ estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty$-type condition, then $\partial Ω$ is $n$-uniformly rectifiable. In the process we obtain a version of Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.

math.CA↗
arXiv2016

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

Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "$S<N$" estimates, and another in terms of a suitable corona decomposition involving harmonic measure.

math.CA↗
arXiv2016

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

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 rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.

math.CA↗
arXiv2016

A characterization of $1$-rectifiable doubling measures with connected supports

Garnett, Killip, and Schul have exhibited a doubling measure $μ$ with support equal to $\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $Γ_{i}$ of finite length for which $μ(\mathbb{R}^{d}\backslash \bigcup Γ_{i})=0$. In this note, we characterize when a doubling measure $μ$ with support equal to a connected metric space $X$ has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $μ$-measure zero with the set of $x\in X$ for which $\liminf_{r\rightarrow 0} μ(B_{X}(x,r))/r>0$.

math.MG↗
arXiv2016

Absolute continuity of harmonic measure for domains with lower regular boundaries

We study absolute continuity of harmonic measure with respect to surface measure on domains $Ω$ that have large complements. We show that if $Γ\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $ω_Ω\ll \mathscr{H}^{d}$ on $Γ\cap \partialΩ$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $Γ\cap\partialΩ$ is a cone point if $Γ$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. Finally, we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.

math.CA↗
arXiv2016

Harmonic measure and Riesz transform in uniform and general domains

Let $Ω\subsetneq\mathbb R^{n+1}$ be open and let $μ$ be some measure supported on $\partialΩ$ such that $μ(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $Ω$ satisfies some scale invariant $A_\infty$ type conditions with respect to $μ$, then the $n$-dimensional Riesz transform $$R_μf(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,dμ(y)$$ is bounded in $L^2(μ)$. We do not assume any doubling condition on $μ$. We also consider the particular case when $Ω$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.

math.CA↗
arXiv2016

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

We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local $Tb$ theorems. The setting is new: we consider conical square functions with cones $\{x \in \mathbb{R}^n \setminus E: |x-y| < 2 \operatorname{dist}(x,E)\}$, $y \in E$, defined on general closed subsets $E \subset \mathbb{R}^n$ supporting a non-homogeneous measure $μ$. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls $B \subset E$, which are doubling and of small boundary. Due to the general set $E$ we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local $Tb$ theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.

math.CA↗
arXiv2016

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

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 this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $Ω\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partialΩ)<\infty$ and a subset $E\subset \partial Ω$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.

math.CA↗
arXiv2016

Tangent measures and absolute continuity of harmonic measure

We show that for uniform domains $Ω\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $α$-dimensional Hausdorff measure unless $α\leq d$. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.

math.CA↗
arXiv2016

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

We show that if $Ω\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(σ)$, where $σ$ is the surface measure of $Ω$, then the outer unit normal to $\partialΩ$ belongs to $\textrm{VMO}(σ)$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.

math.CA↗
arXiv2015

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

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 assumption of Hytönen-Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen-Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures $μ$ in $\mathbb{R}^d$ satisfying $μ(B(x,r)) \le Cr^n$, $n \in (0, d]$. The range of exponents in the Cotlar type inequality depend on $n$. Without assuming buffer we get the full range of exponents $p,q \in (1,2]$ for measures with $n \le 1$, and in general we get $p, q \in [2-ε(n), 2]$, $ε(n) > 0$. Consequences for (non-homogeneous) local $Tb$ theorems are discussed.

math.CA↗
arXiv2015

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.

math.CA↗
arXiv2015

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

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 value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.

math.CA↗
arXiv2015

Uniform domains with rectifiable boundaries and harmonic measure

We assume that $Ω\subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial Ω}$ is locally finite, then the Hausdorff measure $\mathcal{H}^d$ is absolutely continuous with respect to the harmonic measure $ω$ on $\partial Ω$, apart from a set of $\mathcal{H}^d$-measure zero.

math.CA↗