Painlevé's problem and the semiadditivity of analytic capacity
TL;DR: In this article, it was shown that the analytic capacity of a compact set of positive measures can be characterized in terms of the curvature of the measures, and the authors deduced that Θ(E) is semiadditive.
Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.
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TL;DR: A new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author are included.
Abstract: A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.
255 citations
Posted Content•
TL;DR: In this paper, the authors extend these constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces, and illustrate the usefulness of these constructs with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.
Abstract: A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.
151 citations
TL;DR: In this paper, a new class of metric measure spaces is introduced and studied, and Tolsa's space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.
Abstract: A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces (Rn, μ) with μ(B(α, r)) ≤ Crd, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa’s space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.
130 citations
Book•
30 Jan 2017TL;DR: Central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem.
Abstract: This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.
118 citations
TL;DR: In this article, it was shown that the Radon measure is n-rectifiable if the Jones' square function is finite and the Menger curvature of the measure is constant.
Abstract: We show that a Radon measure $${\mu}$$
in $${\mathbb{R}^d}$$
which is absolutely continuous with respect to the n-dimensional Hausdorff measure $${\mathcal{H}^n}$$
is n-rectifiable if the so called Jones’ square function is finite $${\mu}$$
-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to $${\mathcal{H}^{n}}$$
. Further, in this paper we also investigate the relationship between the Jones’ square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.
112 citations
References
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28 Apr 1995TL;DR: In this article, the Fourier transform and its application in general measure theory are discussed. But the authors focus on the analysis of general measures in the complex plane and do not address the problem of analytic capacity in complex planes.
Abstract: Acknowledgements Basic notation Introduction 1 General measure theory 2 Covering and differentiation 3 Invariant measures 4 Hausdorff measures and dimension 5 Other measures and dimensions 6 Density theorems for Hausdorff and packing measures 7 Lipschitz maps 8 Energies, capacities and subsets of finite measure 9 Orthogonal projections 10 Intersections with planes 11 Local structure of s-dimensional sets and measures 12 The Fourier transform and its applications 13 Intersections of general sets 14 Tangent measures and densities 15 Rectifiable sets and approximate tangent planes 16 Rectifiability, weak linear approximation and tangent measures 17 Rectifiability and densities 18 Rectifiability and orthogonal projections 19 Rectifiability and othogonal projections 19 Rectifiability and analytic capacity in the complex plane 20 Rectifiability and singular intervals References List of notation Index of terminology
1,262 citations
842 citations
TL;DR: In this article, the Calderon-Zygmund operator T was used to define the Bilinear form for Lipschitz functions and for smooth functions on smooth functions.
Abstract: 0 Introduction: main objects and results 3 0.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 An application of T1-heorem: electric intensity capacity . . . . . . . . . . . . 7 0.3 How to interpret Calderon–Zygmund operator T? . . . . . . . . . . . . . . . 9 0.3.1 Bilinear form is defined on Lipschitz functions . . . . . . . . . . . . . 10 0.3.2 Bilinear form is defined for smooth functions . . . . . . . . . . . . . . 11 0.3.3 Apriori boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.4 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
394 citations
TL;DR: In this article, the authors give a necessary and sufficient condition for a given set K to lie in a rectifiable curve, which is the image of a finite interval under a Lipschitz mapping.
Abstract: Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image of a finite interval under a Lipschitz mapping. Recall that for a connected set F c C, F is a rectifiable curve (not necessarily simple) if and only if l(F) < ~ , where l(-) denotes one dimensional Hausdorff measure. This classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge (but not necessarily each vertex!) at most twice. If K is a finite set then we are essentially reduced to the classical Traveling Salesman Problem (TSP): Compute the length of the shortest Hami l ton ian cycle which hits all points of K. This is the same, up to a constant multiple, as asking for the inf imum of l(F) where F is a curve, K c F. (Such a F is called a spanning tree in TSP theory.) For infinite sets K, we cannot hope in general to have K be a subset of a Jordan curve. What we should therefore look at is connected sets which conta in K. Let Fmi n be the shortest (minimal) spanning tree. Then we cannot possibly solve our problem for sets K of infinite cardinality if we cannot find F, I(F) < C O/(Fmin) , for any finite set K. (Here and throughout the paper C, Co, C1, c o , etc. denote various universal constants.) While there are several algorithms for computing l(Fml.), these algorithms work for finite graphs satisfying the triangle inequality, and do not use the Euclidean properties of K. (See [13] for an excellent discussion of some of these algorithms.) Therefore these methods cannot solve our problem for general infinite K. We present a method which is a minor modification of a well-known algorithm ("Farthest Insert ion" see [13]) which yields a F with I(F) < C O l(Fmi,). The Farthest Insert ion algorithm has been extensively studied with large numerical calculations on computers, and is experimentally good in the sense that the F produced satisfy I(F) < C O l(F,,,i,) for all examples which have
342 citations