arXiv subjects
Publications and source records attributed to Mihalis Mourgoglou.
We show that if $n\geq 1$, $Ω\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partialΩ$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $ω$ is absolutely continuous with respect to $H^{n}$, then $ω|_E$ is concentrated on an $n$-rectifiable set.
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 $α$-numbers of Tolsa.
We continue the study of local $Tb$ theorems for square functions defined in the upper half-space $(\mathbb{R}^{n+1}_+, μ\times dt/t)$. Here $μ$ is allowed to be a non-homogeneous measure in $\mathbb{R}^n$. In this paper we prove a boundedness result assuming local $L^q$ type testing conditions in the difficult range $q \in (1,2)$. Our theorem is a non-homogeneous version of a result of S. Hofmann valid for the Lebesgue measure. It is also an extension of the recent results of M. Lacey and the first named author where non-homogeneous local $L^2$ testing conditions have been considered.
We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in $L^p$, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for $L$, with data in $L^p(\mathbb{R}^n)$, $BMO(\mathbb{R}^n)$, and $C^α(\mathbb{R}^n)$, and solvability of the Neumann and Regularity problems, with data in the spaces $L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$ and $L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$ respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials in for the $t$-independent operator $L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$.
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 develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces, interpolation and duality. In the way, we reprove the Regularity-Dirichlet duality principle between dual systems and extend it to $H^1-BMO$. We also exhibit and use a similar Neumann-Neumann duality principle.
We continue developing the theory of conical and vertical square functions on $R^{n}$, where $μ$ is a power bounded measure, possibly non-doubling. We provide new boundedness criteria and construct various counterexamples. First, we prove a general local $Tb$ theorem with tent space $T^{2,\infty}$ type testing conditions to characterise the $L^{2}$ boundedness. Second, we completely answer the question, whether the boundedness of our operators on $L^{2}$ implies boundedness on other $L^{p}$ spaces, including the endpoints. For the conical square function, the answers are generally affirmative, but the vertical square function can be unbounded on $L^{p}$ for $p > 2$, even if $μ= dx$. For this, we present a counterexample. Our kernels $s_t$, $t > 0$, do not necessarily satisfy any continuity in the first variable -- a point of technical importance throughout the paper. Third, we construct a non-doubling Cantor-type measure and an associated conical square function operator, whose $L^{2}$ boundedness depends on the exact aperture of the cone used in the definition. Thus, in the non-homogeneous world, the 'change of aperture' technique -- widely used in classical tent space literature -- is not available. Fourth, we establish the sharp $A_{p}$-weighted bound for the conical square function under the assumption that $μ$ is doubling.
In this article we prove solvability results for $L^2$ boundary value problems of some elliptic systems $Lu=0$ on the upper half-space $\R^{n+1}_{+}, n\ge 1$, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.
We characterize the boundedness of square functions in the upper half-space with general measures. The short proof is based on an averaging identity over good Whitney regions.
We prove a local Tb Theorem for square functions, in which we assume L^p control of the pseudo-accretive system, with p>1 extending the work of S. Hofmann to domains with Ahlfors-David regular boundaries.
Given $E \subset {\Bbb R}^d$, define the \emph{volume set} of $E$, ${\mathcal V}(E)= \{det(x^1, x^2, ... x^d): x^j \in E\}$. In $\R^3$, we prove that ${\mathcal V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of $E\subset \Bbb R^3$ is greater than 13/5, or $E$ is a product set of the form $E=B_1\times B_2\times B_3$ with $B_j\subset\R,\, dim_{\mathcal H}(B_j)>2/3,\, j=1,2,3$. We show that the same conclusion holds for $\V(E)$ of Salem subsets $E\subset\R^d$ with $\hde>d-1$, and give applications to discrete combinatorial geometry.
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 Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in (\cite{HKKMMS10}). We also obtain new upper bounds for the number of times an angle can occur among $N$ points in ${\mathbb R}^d$, $d \ge 4$, motivated by the results of Apfelbaum and Sharir (\cite{AS05}) and Pach and Sharir (\cite{PS92}). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.
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 $Δ(E)=\{|x-y|: x,y \in E \}$ contains an interval. We prove this result for distance sets $Δ_B(E)=\{{||x-y||}_B: x,y \in E \}$, where ${|| \cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.
Given $E \subset \mathbb{R}^d$, $d \ge 2$, define ${\mathcal D}(E) \equiv {(x-y)/|x-y|: x,y \in E} \subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then $σ({\mathcal D}(E))>0$, where $σ$ denotes the surface measure on $S^{d-1}$. This result is sharp since the conclusion fails to hold if $E$ is a $(d-1)$-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir (\cite{PPS04}, \cite{PPS07}) on angles determined by finite subsets of $\mathbb{R}^d$. Also define ${\mathcal A}(E)=\{θ(x,y,z): x,y,z \in E\},$ where $θ(x,y,z)$ is the angle between $x-y$ and $y-z$. We use the techniques developed to handle the problem of directions and results on distance sets previously obtained by Wolff and Erdogan to prove that if the Hasudorff dimension of $E$ is greater than $(d-1)/2+1/3$, then the Lebesgue measure of ${\mathcal A}(E)$ is positive. This result can be viewed as a continuous analog of a recent result of Apfelbaum and Sharir (\cite{AS05}). At the end of this paper we show that our continuous results can be used to recover and in some case improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set $P \subset {\Bbb R}^d$, $d \ge 3$, satisfying a certain discrete energy condition (Definition \ref{adaptablemama}), determines $\gtrapprox # P$ distinct directions and $\gtrapprox {(# P)}^{6/(3d-1)}$ distinct angles. In two dimensions, the lower bound on the number of angles is $\gtrapprox # P$.