Curriculum Vitae Xavier Tolsa Dom`Enech June 30, 2021
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A Measure on the Harmonic Boundary of a Riemann Surface
A MEASURE ON THE HARMONIC BOUNDARY OF A RIEMANN SURFACE MITSURU NAKAI Introduction 1. In the usual theory of harmonic functions on a plane domain, the fact that the boundary of the domain is realized relative to the complex plane plays an essential role and supplies many powerful tools, for instance, the solution of Dirichlet problem. But in the theory of harmonic functions on a general domain, i.e. on a Riemann surface, the main difficulty arises from the lack of the "visual" boundary of the surface. Needless to say, in general we cannot expect to get the "relative" boundary with respect to some other larger surface. In view of this, we need some "abstract" compactification. It seems likely that we cannot expect to get the "universal" boundary which is appropriate for any harmonic functions since there exist many surfaces which do not admit some classes of harmonic functions as the classification theory shows. Hence we need many compactifications corresponding to what class of harmonic functions we are going to investigate. There are two typical compactifications, as Royden [10] pointed out, Martin's compactification and Royden's compactification. The former seems appropriate for the study of //"^-functions and the latter aims to be used for the study of HBD-ίunctions, which we are going to investigate in this paper. Λ similar investigation is carried out by Kuramochi in the direction of Martin. 2. Royden's compactification is first introduced by Royden and has been studied by Mori, Mori-Ota, Kusunoki-Mori and the present author. But these investigations seem to be restricted for ffi?D-functions. -
A Very Short Survey on Boundary Behavior of Harmonic Functions
A VERY SHORT SURVEY ON BOUNDARY BEHAVIOR OF HARMONIC FUNCTIONS BLACK Abstract. This short expository paper aims to use Dirichlet boundary value problem to elab- orate on some of the interactions between complex analysis, potential theory, and harmonic analysis. In particular, I will outline Wiener's solution to the Dirichlet problem for a general planar domain using harmonic measure and prove some elementary results for Hardy spaces. 1. Introduction Definition 1.1. Let Ω Ă R2 be an open set. A function u P C2pΩ; Rq is called harmonic if B2u B2u (1.1) ∆u “ ` “ 0 on Ω: Bx2 By2 The notion of harmonic function can be generalized to any finite dimensional Euclidean space (or on (pseudo)Riemannian manifold), but the theory enjoys a qualitative difference in the planar case due to its relation to the magic properties of functions of one complex variable. Think of Ω Ă C (in this paper I will use R2 and C interchangeably when there is no confusion). Then the Laplace operator takes the form B B B 1 B B B 1 B B (1.2) ∆ “ 4 ; where “ ´ i and “ ` i : Bz¯ Bz Bz 2 Bx By Bz¯ 2 Bx By ˆ ˙ ˆ ˙ Let HolpΩq denote the space of holomorphic functions on Ω. It is easy to see that if f P HolpΩq, then both <f and =f are harmonic functions. Conversely, if u is harmonic on Ω and Ω is simply connected, we can construct a harmonic functionu ~, called the harmonic conjugate of u, via Hilbert transform (or more generally, B¨acklund transform), such that f “ u ` iu~ P HolpΩq. -
Harmonic Measure
Chapter 6 Harmonic Measure Prologue: It is unfortunate that most graduate com- plex analysis courses do not treat harmonic measure. Indeed, most complex analysis texts do not cover the topic. The term “harmonic measure” was introduced by R. Nevanlinna in the 1920s. But the ideas were antici- pated in work of the Riesz brothers and others. It is still studied intensively today. Harmonic measure is a natural outgrowth of the study of the Dirichlet problem. It is a decisive tool for mea- suring the growth of functions, and to seeing how they bend. It has been used in such diverse applications as the corona problem and in mapping problems. Harmonic measure has particularly interesting applications in prob- ability theory—especially in the consideration of diffus- tions and Brownian motion. It requires a little sophistication to appreciate har- monic measure. And calculating the measure—working through the examples—requires some elbow grease. But the effort is worth it and the results are substantial. This chapter gives a basic introduction to the idea of harmonic measure. 169 79344-3_Printer Proof.pdf 183 “CH06” — 2015/10/26 — 17:20 — page 169 — #1 11/6/2015 1:52:49 PM 170 CHAPTER 6. HARMONIC MEASURE 6.1 The Idea of Harmonic Measure Capsule: Harmonic measure is a way of manipulating the data that comes from the Dirichlet problem. It is a way of measuring the geom- etry of a domain, and can be used to control the growth of harmonic functions. Let Ω ⊆ C be a domain with boundary consisting of finitely many Jordan curves (we call such a domain a Jordan domain). -
Nontangential Limits in Pt(Μ)-Spaces and the Index of Invariant Subspaces
Annals of Mathematics, 169 (2009), 449{490 Nontangential limits in Pt(µ)-spaces and the index of invariant subspaces By Alexandru Aleman, Stefan Richter, and Carl Sundberg* Abstract Let µ be a finite positive measure on the closed disk D in the complex plane, let 1 ≤ t < 1, and let P t(µ) denote the closure of the analytic polyno- t mials in L (µ). We suppose that D is the set of analytic bounded point evalua- tions for P t(µ), and that P t(µ) contains no nontrivial characteristic functions. It is then known that the restriction of µ to @D must be of the form hjdzj. We prove that every function f 2 P t(µ) has nontangential limits at hjdzj-almost every point of @D, and the resulting boundary function agrees with f as an element of Lt(hjdzj). Our proof combines methods from James E. Thomson's proof of the ex- istence of bounded point evaluations for P t(µ) whenever P t(µ) 6= Lt(µ) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure ν in the complex plane we are able to describe locations of bounded point evaluations for P t(ν) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < 1 dim M=zM = 1 for every nonzero invariant subspace M of P t(µ) if and only if h 6= 0. -
Dimensional Properties of the Harmonic Measure for a Random Walk on a Hyperbolic Group Vincent Le Prince
Dimensional properties of the harmonic measure for a random walk on a hyperbolic group Vincent Le Prince To cite this version: Vincent Le Prince. Dimensional properties of the harmonic measure for a random walk on a hyperbolic group. Transactions of the American Mathematical Society, American Mathematical Society, 2007, 359 (6), pp.2881-2898. hal-00003284 HAL Id: hal-00003284 https://hal.archives-ouvertes.fr/hal-00003284 Submitted on 15 Nov 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Dimensional properties of the harmonic measure for a random walk on a hyperbolic group Vincent Le Prince 15th November 2004 Abstract: This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure ν associated with such a random walk. We first establish a link of the form dim ν ≤ h/l between the dimension of the harmonic measure, the asymptotic entropy h of the random walk and its rate of escape l. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure. -
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Arxiv:1904.13116V2 [Math.AP] 13 Jul 2020 Iosfudto Rn 696 SM
TRANSFERENCE OF SCALE-INVARIANT ESTIMATES FROM LIPSCHITZ TO NON-TANGENTIALLY ACCESSIBLE TO UNIFORMLY RECTIFIABLE DOMAINS STEVE HOFMANN, JOSE´ MARIA´ MARTELL, AND SVITLANA MAYBORODA Abstract. In relatively nice geometric settings, in particular, on Lipschitz domains, absolute con- tinuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to ε-approximability, for solutions to the second order divergence form elliptic partial differential equations Lu = div (A u) = 0. In more general situa- tions, notably, in an open set Ω with a uniformly rectifiable boundary,− ∇ absolute continuity of elliptic measure with respect to the surface measure may fail, already for the Laplacian. In the present paper, the authors demonstrate that nonetheless, Carleson measure estimates, square function estimates, and ε-approximability remain valid in such Ω, for solutions of Lu = 0, provided that such solutions enjoy these properties in Lipschitz subdomains of Ω. Moreover, we establish a general real-variable transference principle, from Lipschitz to chord-arc domains, and from chord-arc to open sets with uniformly rectifiable boundary, that is not restricted to harmonic functions or even to solutions of elliptic equations. In particular, this allows one to deduce the first Carleson measure estimates and square function bounds for higher order systems on open sets with uniformly rectifiable boundaries and to treat subsolutions and subharmonic functions. Contents 1. Introduction 2 2. Preliminaries 8 2.1. Case ADR 16 2.2. Case UR 16 2.3. Case 1-sided CAD 19 2.4. Case CAD 20 2.5. Some important notation 20 3. -
How Twisted Can a Jordan Curve Be?
HOW TWISTED CAN A JORDAN CURVE BE? Manfred Sauter Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless num- ber of possible exceptions. — Felix Klein (1849–1925) We start by fixing notation. A closed curve is a continuous map γ : [0, 1] → 2 R with γ(0) = γ(1), and it is a closed Jordan curve if in addition γ|[0,1) is injective. Similarly one defines a (Jordan) arc, a polygonal (Jordan) arc and a closed polygonal (Jordan) curve. It is well-known, of course, that a closed Jordan curve γ partitions the plane into three connected components: the bounded open interior int γ, the unbounded open exterior ext γ and the compact (image of the) curve itself, which is the common boundary of the first two components. For a captivating historical overview of the struggle to understand curves and toplological dimension we refer to [Cri99], where most of the classical topological results that appear in this note are featured. For a closed polygonal Jordan curve γ there is a simple criterion to check whether a point x ∈ R2 \ γ is in its interior. This criterion is algorithmic in nature and useful in computational geometry, see for example [O’R98, Section 7.4]. Consider a ray emerging from x towards infinity and count the number of intersections with γ. If the ray is chosen such that it intersects the segments of γ only ouside of their end points (which can be easily arranged for a polygonal Jordan curve) or if intersections with the end points are counted appropriately, the point x is in the interior of γ if and only if the number of intersections is odd. -
An Improved Riemann Mapping Theorem and Complexity In
AN IMPROVED RIEMANN MAPPING THEOREM AND COMPLEXITY IN POTENTIAL THEORY STEVEN R. BELL Abstract. We discuss applications of an improvement on the Riemann map- ping theorem which replaces the unit disc by another “double quadrature do- main,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann Mapping Theo- rem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature do- main close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains. 1. Introduction The unit disc is the most famous example of a “double quadrature domain.” The average of an analytic function on the disc with respect to both area measure and with respect to boundary arc length measure yield the value of the function at the origin when these averages make sense. The Riemann Mapping Theorem states that when Ω is a simply connected domain in the plane that is not equal to the whole complex plane, a biholomorphic map of Ω to this famous double quadrature domain exists. We proved a variant of the Riemann Mapping Theorem in [16] that allows the domain Ω = C to be simply or finitely connected. -
Carleson Measures and Elliptic Boundary Value Problems 3
1 Carleson measures and elliptic boundary value 2 problems 3 4 Jill Pipher 5 Abstract. In this article, we highlight the role of Carleson measures in elliptic boundary value prob- 6 lems, and discuss some recent results in this theory. The focus here is on the Dirichlet problem, with 7 measurable data, for second order elliptic operators in divergence form. We illustrate, through selected 8 examples, the various ways Carleson measures arise in characterizing those classes of operators for 9 which Dirichlet problems are solvable with classical non-tangential maximal function estimates. 10 Mathematics Subject Classification (2010). Primary 42B99, 42B25, 35J25, 42B20. 11 Keywords. Carleson measures, elliptic divergence form equations, A1, boundary value problems. 12 1. Introduction 13 Measures of Carleson type were introduced by L. Carleson in [9] and [10] to solve a problem 14 in analytic interpolation, via a formulation that exploited the duality between Carleson mea- 15 sures and non-tangential maximal functions (defined below). Carleson measures have since 16 become one of the most important tools in harmonic analysis, playing a fundamental role in 17 the study of singular integral operators in particular, through their connection with BMO, 18 the John-Nirenberg space of functions of bounded mean oscillation. We aim to describe, 19 through some specific examples, the ubiquitous role of measures of this type in the theory 20 of boundary value problems, especially with regard to sharp regularity of “elliptic” measure, 21 the probability measure arising in the Dirichlet problem for second order divergence form 22 elliptic operators. Perhaps the first connection between Carleson measures and boundary 23 value problems was observed by C. -
Who Is Lennart Carleson?
The Abel Prize for 2006 has been awarded Lennart Carleson, The Royal Institute of Technology, Stockholm, Sweden Professor Lennart Carleson will accept the Abel Prize for 2006 from His Majesty King Harald in a ceremony at the University of Oslo Aula, at 2:00 p.m. on Tuesday, 23 May 2006. In this background paper, we shall give a description of Carleson and his work. This description includes precise discussions of his most important results and attempts at popular presentations of those same results. In addition, we present the committee's reasons for choosing the winner and Carleson's personal curriculum vitae. This paper has been written by the Abel Prize's mathematics spokesman, Arne B. Sletsjøe, and is based on the Abel Committee's deliberations and prior discussion, relevant technical literature and discussions with members of the Abel Committee. All of this material is available for use by the media, either directly or in an adapted version. Oslo, Norway, 23 March 2006 Contents Introduction .................................................................................................................... 2 Who is Lennart Carleson? ............................................................................................... 3 Why has he been awarded the Abel Prize for 2006? ........................................................ 4 Popular presentations of Carleson's results ...................................................................... 6 Convergence of Fourier series.................................................................................... -
Orthogonal Functions in H∞
Pacific Journal of Mathematics ORTHOGONAL FUNCTIONS IN H∞ CHRISTOPHER J. BISHOP Volume 220 No. 1 May 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 220, No. 1, 2005 ORTHOGONAL FUNCTIONS IN H∞ CHRISTOPHER J. BISHOP We construct examples of H∞ functions f on the unit disk such that the push-forward of Lebesgue measure on the circle is a radially symmetric measure µf in the plane, and we characterize which symmetric measures can occur in this way. Such functions have the property that { f n} is orthog- onal in H2, and provide counterexamples to a conjecture of W. Rudin, in- dependently disproved by Carl Sundberg. Among the consequences is that there is an f in the unit ball of H∞ such that the corresponding composition operator maps the Bergman space isometrically into a closed subspace of the Hardy space. 1. Introduction Let H ∞ denote the algebra of bounded holomorphic functions on the unit disk ބ, ∞ ∞ let ᐁ be the closed unit ball of H and let ᐁ0 = { f ∈ ᐁ : f (0) = 0}. If f ∈ H then it has radial boundary values (which we also call f ) almost everywhere on the unit circle ޔ. We say that f is orthogonal if the sequence of powers { f n : n = 0, 1,... } is orthogonal, that is, if Z f n f¯m dθ = 0 ޔ whenever n 6= m. In this paper we will characterize orthogonal functions in H ∞ −1 in terms of the Borel probability measure µf (E) = | f (E)|, where | · | denotes Lebesgue measure on ޔ, normalized to have mass 1. We will also determine exactly which measures arise in this way. -
HARMONIC MEASURE, TRUE TREES and QUASICONFORMAL FOLDING Christopher Bishop, Stony Brook International Congress of Mathematicians
HARMONIC MEASURE, TRUE TREES AND QUASICONFORMAL FOLDING Christopher Bishop, Stony Brook International Congress of Mathematicians August 6, 2018 Rio de Janeiro, Brazil www.math.sunysb.edu/~bishop/lectures Harmonic measure = hitting distribution of Brownian motion Suppose Ω is a planar Jordan domain. Harmonic measure = hitting distribution of Brownian motion Let E be a subset of the boundary, ∂Ω. Harmonic measure = hitting distribution of Brownian motion Choose an interior point z ∈ Ω. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) = probability a particle started at z first hits ∂Ω in E. Harmonic measure = hitting distribution of Brownian motion ω(z,E, Ω) ≈ 64/100. What if we move starting point z? Harmonic measure = hitting distribution of Brownian motion Different z gives ω(z,E, Ω) ≈ 12/100.