Related papers: Dimensions of Riesz products and pluriharmonic mea…
In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral…
The decay rate of Riesz capacity as the exponent increases to the dimension of the set is shown to yield Hausdorff measure. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For…
We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity…
In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by…
We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold $M$. Applications are given to the Hausdorff measure of the intersection of the variety with $M$…
We provide an estimate from below for the lower Hausdorff dimension of measures on the unit circle based on the arithmetic properties of their spectra. We obtain our bounds via application of a general result for abstract $q$-regular…
A metric space $\mathcal{S}$ is called a \defn{quasisphere} if there is a quasisymmetric homeomorphism $f\colon S^2\to \mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$.…
Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^s$ is the unique Borel probability measure with support in $A$ that minimizes $$…
Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present…
Basic properties of Hausdorff content, dimension, and measure of subsets of metric spaces are discussed, especially in connection with Lipschitz mappings and topological dimension.
In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis to the…
These notes deal with metric spaces, Hausdorff measures and dimensions, Lipschitz mappings, and related topics. The reader is assumed to have some familiarity with basic analysis, which is also reviewed.
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the…
These are the lecture notes of a seminar held at the Universitat Aut\`onoma de Barcelona where the Jones-Wolff theorem about the dimension of harmonic measure in the plane is explained in full detail, for non-expert readers.
The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to…
Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna…
We consider Non Autonomous Conformal Iterative Function Systems (NACIFS) and their limit set. Our main concern is harmonic measure and its dimensions : Hausdorff and Packing. We prove that this two dimensions are continuous under…
In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff…
We discuss on some families of skew product maps on a square. For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor. And we prove that there exists an ergodic measure with full…
The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some…