DOCSLIB.ORG

Moduli and Domains

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Moduli and Domains Springer Monographs in Mathematics For other titles published in this series, go to http://www.springer.com/series/3733 Olli Martio • Vladimir Ryazanov • Uri Srebro • Eduard Yakubov Moduli in Modern Mapping Theory With 12 Illustrations 123 Olli Martio Vladimir Ryazanov University of Helsinki Institute of Applied Mathematics Helsinki and Mechanics of National Academy Finland of Sciences of Ukraine olli.martio@helsinki.fi Donetsk Ukraine [email protected] Uri Srebro Eduard Yakubov Technion - Israel Institute H.I.T. - Holon Institute of Technology of Technology Holon Haifa Israel Israel [email protected] [email protected] ISSN: 1439-7382 ISBN: 978-0-387-85586-8 e-ISBN: 978-0-387-85588-2 DOI 10.1007/978-0-387-85588-2 Library of Congress Control Number: 2008939873 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec- tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com Dedicated to 100 Years of Lars Ahlfors Preface The purpose of this book is to present modern developments and applications of the techniques of modulus or extremal length of path families in the study of map- pings in Rn, n ≥ 2, and in metric spaces. The modulus method was initiated by Lars Ahlfors and Arne Beurling to study conformal mappings. Later this method was extended and enhanced by several other authors. The techniques are geomet- ric and have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on rather recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs. Helsinki O. Martio Donetsk V. Ryazanov Haifa U. Srebro Holon E. Yakubov 2007 Contents 1 Introduction and Notation ...................................... 1 2 Moduli and Capacity ........................................... 7 2.1 Introduction . ............................................ 7 2.2 Moduli in Metric Spaces. ................................. 7 2.3 Conformal Modulus ........................................ 11 2.4 Geometric Definition for Quasiconformality . ............... 13 2.5 Modulus Estimates . ........................................ 14 2.6 Upper Gradients and ACCp Functions . ...................... 17 n 2.7 ACCp Functions in R and Capacity . ...................... 21 2.8 Linear Dilatation . ........................................ 25 2.9 Analytic Definition for Quasiconformality . ............... 31 2.10 Rn as a Loewner Space . ................................. 34 2.11Quasisymmetry............................................ 40 3 Moduli and Domains ........................................... 47 3.1 Introduction . ............................................ 47 3.2 QED Exceptional Sets . ................................. 48 3.3 QED Domains and Their Properties . ...................... 52 3.4 UniformandQuasicircleDomains............................ 55 3.5 Extension of Quasiconformal and Quasi-Isometric Maps . .... 62 3.6 Extension of Local Quasi-Isometries . ...................... 69 3.7 Quasicircle Domains and Conformal Mappings . ............... 71 3.8 On Weakly Flat and Strongly Accessible Boundaries . ........... 73 ∈ 1 4 Q-Homeomorphisms with Q Lloc ............................... 81 4.1 Introduction . ............................................ 81 4.2 Examples of Q-homeomorphisms............................. 82 n−1 4.3 Differentiability and KO(x, f ) ≤ Cn Q (x) a.e................. 83 1,1 4.4 Absolute Continuity on Lines and Wloc ........................ 86 4.5 LowerEstimateofDistortion................................. 89 x Contents 4.6 Removal of Singularities . ................................. 90 4.7 Boundary Behavior . ........................................ 91 4.8 Mapping Problems . ........................................ 92 5 Q-homeomorphisms with Q in BMO ............................. 93 5.1 Introduction . ............................................ 93 5.2 MainLemmaonBMO...................................... 94 5.3 Upper Estimate of Distortion ................................. 96 5.4 Removal of Isolated Singularities ............................. 97 5.5 On Boundary Correspondence . ............................. 97 5.6 Mapping Problems . ........................................ 99 5.7 SomeExamples............................................101 6 More General Q-Homeomorphisms ..............................103 6.1 Introduction . ............................................103 6.2 Lemma on Finite Mean Oscillation . ......................104 6.3 On Super Q-Homeomorphisms...............................108 6.4 Removal of Isolated Singularities .............................109 6.5 Topological Lemmas . .................................114 6.6 On Singular Sets of Length Zero ..............................118 6.7 Main Lemma on Extension to Boundary . ......................121 6.8 Consequences for Quasiextremal Distance Domains . ...........123 6.9 On Singular Null Sets for Extremal Distances ...................125 6.10 Applications to Mappings in Sobolev Classes . ...............126 7RingQ-Homeomorphisms .......................................131 7.1 Introduction . ............................................131 7.2 On Normal Families of Maps in Metric Spaces . ...............132 7.3 Characterization of Ring Q-Homeomorphisms..................135 7.4 EstimatesofDistortion......................................137 7.5 On Normal Families of Ring Q-Homeomorphisms...............141 7.6 On Strong Ring Q-Homeomorphisms..........................142 8 Mappings with Finite Length Distortion (FLD) ....................145 8.1 Introduction . ............................................145 8.2 Moduli of Cuttings and Extensive Moduli ......................147 8.3 FMD Mappings ............................................149 8.4 FLD Mappings ............................................152 8.5 Uniqueness Theorem . .................................154 8.6 FLD and Q-Mappings . .................................156 8.7 OnFLDHomeomorphisms..................................159 8.8 OnSemicontinuityofOuterDilatations........................164 8.9 On Convergence of Matrix Dilatations . ......................169 8.10 Examples and Subclasses . .................................172 Contents xi 9LowerQ-Homeomorphisms .....................................175 9.1 Introduction . ............................................175 9.2 On Moduli of Families of Surfaces . ......................176 9.3 Characterization of Lower Q-Homeomorphisms.................180 9.4 EstimatesofDistortion......................................183 9.5 Removal of Isolated Singularities .............................184 9.6 On Continuous Extension to Boundary Points ...................185 9.7 OnOneCorollaryforQEDDomains..........................186 9.8 On Singular Null Sets for Extremal Distances ...................186 9.9 LemmaonClusterSets......................................187 9.10 On Homeomorphic Extensions to Boundaries . ...............190 10 Mappings with Finite Area Distortion ............................193 10.1 Introduction . ............................................193 10.2 Upper Estimates of Moduli . .................................194 10.3 On Lower Estimates of Moduli . .............................198 10.4 Removal of isolated singularities ..............................199 10.5 Extension to Boundaries . .................................200 10.6 Finitely Bi-Lipschitz Mappings . .............................202 11 On Ring Solutions of the Beltrami Equation .......................205 11.1 Introduction . ............................................205 11.2 Finite Mean Oscillation . .................................207 11.3 Ring Q-HomeomorphismsinthePlane ........................211 11.4DistortionEstimates........................................216 11.5 General Existence Lemma and Its Corollaries . ...............224 11.6 Representation, Factorization and Uniqueness Theorems . ....228 11.7Examples.................................................232 12 Homeomorphisms with Finite Mean Dilatations ...................237 12.1 Introduction . ............................................237 12.2 Mean Inner and Outer Dilatations .............................239 12.3 On Distortion of p-Moduli . .................................242 12.4 Moduli of Surface Families Dominated by Set Functions . ....244 12.5 Alternate Characterizations of Classical Mappings . ...........247 12.6 Mappings (α,β)-Quasiconformal in the Mean . ...............249 12.7 Coefficients of Quasiconformality of Ring Domains . ...........251 13 On Mapping Theory in Metric Spaces ............................257 13.1 Introduction . ............................................257 13.2 Connectedness in Topological Spaces . ......................259 13.3 On Weakly Flat and Strongly Accessible
Load more

Recommended publications
  • Removable Singularities for Planar Quasiregular Mappings
    Removable singularities for planar quasiregular mappings Albert Clop PhD Thesis Memoria` presentada per aspirar al grau de doctor en Matematiques` iii iv CERTIFIQUEM que aquesta memoria` ha estat realitzada per Albert Clop, sota la direccio´ dels Drs. Joan Mateu Bennassar i Joan Orobitg Huguet. Bellaterra, octubre del 2006. Joan Mateu Bennassar Joan Orobitg Huguet v Introduction A compact planar set E is said to be removable for bounded analytic functions if for any open set W ⊃ E, any bounded function f : W ! C holomorphic on W n E admits a holomorphic extension to W. It is enough to care about the case W = C. The Painleve´ problem consists on characterizing these sets in metric and geometric terms. The analytic capacity of a compact set E, introduced by Ahlfors [3], is defined by 0 g(E) = sup j f (¥)j; f 2 H(C n E), k f k¥ ≤ 1, f (¥) = 0 . This set function vanishes only on removable sets. Some excellent introductions to g are given by H. Pajot [43] and J. Verdera [55]. As a matter of fact, H1(E) = 0 implies g(E) = 0, while if dim(E) > 1 then g(E) > 0. Thus, 1 is the critical dimension for the Painleve´ problem. Some easier variants of this problem are obtained when replacing L¥ by other classes < of functions, such as BMO or Lipa for 0 a ≤ 1. To specify: • Kaufman [31] showed that E is removable for BMO analytic functions if and only if M1(E) = 0. • Dolzenkoˇ [16] proved that E is removable for Lipa analytic functions if and only if M1+a(E) = 0.
    [Show full text]
  • P-Laplace Operator, Quasiregular Mappings, and Picard-Type Theorems 1. Introduction
    Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05) p-Laplace operator, quasiregular mappings, and Picard-type theorems Ilkka Holopainen and Pekka Pankka Abstract. We describe the role of p-harmonic functions and forms in the theory of quasiregular mappings. Keywords. Quasiregular mapping, p-harmonic function, p-harmonic form, conformal capacity. 2000 MSC. Primary 58J60, 30C65; Secondary 53C20, 31C12, 35J60. Contents 1. Introduction 69 2. -harmonic functions 72 A 3. Morphism property and its consequences 75 3.1. Sketch of the proof of Reshetnyak's theorem 77 Further properties of f 79 4. Modulus and capacity inequalities 80 5. Liouville-type results for -harmonic functions 82 A 6. Liouville-type results for quasiregular mappings 88 7. Picard-type theorems 89 8. Quasiregular mappings, p-harmonic forms, and de Rham cohomology 97 References 100 1. Introduction In this survey we emphasize the importance of the p-Laplace operator as a tool to prove basic properties of quasiregular mappings, as well as Liouville- and Picard-type results for quasiregular mappings between given Riemannian manifolds. Quasiregular mappings were introduced by Reshetnyak in the mid sixties in a series of papers; see e.g. [36], [37], and [38]. An interest in studying these mappings arises from a question about the existence of a geometric function theory in real dimensions n 3 generalizing the theory of holomorphic functions ≥ C C: ! 70 Ilkka Holopainen and Pekka Pankka IWQCMA05 Definition 1.1. A continuous mapping f : U Rn of a domain U Rn is called quasiregular (or a mapping of bounded distortion! ) if ⊂ 1;n n (1) f Wloc (U; R ); and (2) there2 exists a constant K 1 such that ≥ n f 0(x) K J (x) for a.e.
    [Show full text]
  • Quasiregular Mappings in Even Dimensions
    Acta Math., 170 (1993), 29-81 Quasiregular mappings in even dimensions by TADEUSZ IWANIEC(1) and GAVEN MARTIN Institut Mittag-LeJ~er Institut Mittag-LeJCfler Djursholm, Sweden Djursholm, Sweden To Professor F. W. Gehring on the occasion of his 65th birthday Contents O. Introduction 1. Notation 2. Some exterior algebra 3. Differential forms in Lnm (i2, A n) 4. Differential systems for quasiregular mappings 5. Liouville Theorem in even dimensions 6. Hodge theory in LP(R n) 7. The Beltrami equation in even dimensions 8. The Beurling-Ahlfors operator 9. Regularity theorems for quasiregular mappings 10. The Caccioppoli type estimate 11. Removability theorems for quasiregular mappings 12. Some examples Appendix: The 4-dimensional case 0. Introduction This paper grew out of our study of the paper of S. Donaldson and D. Sullivan "Quasicon- formal 4-Manifolds" [DS]. In that paper the authors develop a quasiconformal Yang-Mills theory by studying elliptic index theory with measurable coefficients in four dimensions via the Calder6n-Zygmund theory of singular integrals. Other approaches to index the- ory on quasiconformal manifolds have been found by N. Teleman [T] and there are related results due to D. Freed and K, Uhlenbeck [FU] and others. We soon realized that there were many other applications of these ideas to the general theory of quasiconformal mappings. In this paper we present just a few of them. (1)Research supported in part by a grant from the U.S. National Science Foundation, DMS-9007946 30 T. IWANIEC AND G. MARTIN A principal feature of our methods is that they are quite explicit and closely mimic the two dimensional theory as developed by Ahlfors [A1], Boyarski [B] and Lehto [Le].
    [Show full text]
  • Arxiv:1807.07683V1 [Math.CV] 20 Jul 2018 Htare That F Differential Ucindfie Ntecmlxplane Complex the on Defined Function Ae Ntecneto Ouu O Ahfmle.Ltrnon- Definitions
    THE RICKMAN-PICARD THEOREM MARIO BONK AND PIETRO POGGI-CORRADINI Dedicated to the memory of Juha Heinonen and Seppo Rickman ABSTRACT. We give a new and conceptually simple proof of the Rickman-Picard theo- rem for quasiregular maps based on potential-theoretic methods. 1. INTRODUCTION The classical Picard theorem in complex analysis states that a non-constant analytic function defined on the complex plane C omits at most one complex value. There are a dozen or so proofs of this theorem using surprisingly diverse and unexpected approaches. In the 1970s, efforts were made to generalize this theorem to quasiregular maps defined on real Euclidean spaces. Although quasiregular maps (defined below) provide a gener- alization of analytic maps, many of the proofs that work in the analytic case fail in the higher-dimensional setting. In 1980 Seppo Rickman [Ri1] was the first to establish an analog of Picard’s theorem for quasiregular maps in higher dimensions. His proof was based on the concept of modulus for path families. Later non-linear potential theory was used to give alternative proofs. To formulate the Rickman-Picard Theorem, we first review some basic definitions. Let M and N be connected and oriented Riemannian n-manifolds, where n ≥ 2. Amap f : M → N is called K-quasiregular, where K ≥ 1, if f has distributional derivatives that are Ln-integrable (with respect to the Riemannian measure on M) and if the formal differential Df(p): TpM → Tf(p)N satisfies kDf(p)kn ≤ K det(Df(p)) for almost every p ∈ M. Here kDf(p)k denotes operator norm of Df(p) with respect to arXiv:1807.07683v1 [math.CV] 20 Jul 2018 the norms on the tangent spaces TpM and Tf(p)N induced by the Riemannian structures on M and N, respectively.
    [Show full text]
  • SMOOTH QUASIREGULAR MAPPINGS with BRANCHING by MARIO BONK and JUHA HEINONEN
    SMOOTH QUASIREGULAR MAPPINGS WITH BRANCHING by MARIO BONK and JUHA HEINONEN ABSTRACT We give an example of a C 3−-smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in n-space has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching. 1. Introduction In this paper, we prove three theorems about quasiregular mappings in space. First we show, contrary to some expectations, that there exist continuously differ- entiable quasiregular mappings in 3-space with nonempty branch set. Then we show that the Hausdorff dimension of the branch set of a quasiregular mapping in a domain in Rn is bounded away from n by a constant that only depends on n and the dilatation of the mapping. Moreover, it turns out that these two prob- lems are related, and we use the Hausdorff dimension estimate to provide a new relation between the smoothness of the mapping and its branching; the example shows that this relation is qualitatively sharp in dimension three. Precise formula- tions of the results will be given momentarily. First we require some notation and terminology. Throughout this paper, G denotes a domain in Rn, n ≥ 2. A continuous and : → n 1,n( ; n) nonconstant mapping f G R in the local Sobolev space Wloc G R is called quasiregular,orK-quasiregular, K ≥ 1,if n (1.1) | f (x)| ≤ KJf (x) for almost every x ∈ G.Heref (x) is the formal differential matrix of f and J f (x) its Jacobian determinant.
    [Show full text]
  • Ahlfors' Contribution to the Theory of Meromorphic
    AHLFORS’ CONTRIBUTION TO THE THEORY OF MEROMORPHIC FUNCTIONS A. Eremenko This article is an exposition for non-specialists of Ahlfors’ work in the theory of meromorphic functions. When the domain is not specified we mean meromorphic functions in the complex plane C. The theory of meromorphic functions probably starts with Briot-Bouquet’s book [17], where the terms “pole”, “essential singularity” and “meromorphic” were in- troduced and what is known now as the Casorati–Weierstrass Theorem was stated for the first time. A major discovery was Picard’s theorem (1879), which says that a meromorphic function omitting three values in the extended complex plane C¯ is constant. The modern theory of meromorphic functions started with attempts to give an “elementary proof” of this theorem. These attempts culminated in R. Nevanlinna’s theory which was published first in 1925. Nevanlinna’s books [45] and [46] were very influential and shaped much of research in function theory in this century. Nevanlinna theory, also known as value distribution theory, was consid- ered one of the most important areas of research in 1930-40, so it is not surprising that Ahlfors started his career with work in this subject (besides he was Nevan- linna’s student). There are two very good sources about this early work of Ahlfors: the first volume of his collected papers [1], where most of the papers have been supplied with Ahlfors’ own commentaries, and Drasin’s article [22]. I used both sources substantially in preparing this lecture, trying to minimize the intersection with Drasin’s survey. 1. Type problem.
    [Show full text]
  • $ N $-Harmonic Coordinates and the Regularity of Conformal Mappings
    Math. Res. Lett. 21 (2014), no. 2, 341–361 c International Press 2014 n-HARMONIC COORDINATES AND THE REGULARITY OF CONFORMAL MAPPINGS Tony Liimatainen and Mikko Salo Abstract. This paper studies the smoothness of conformal mappings between two Rie- mannian manifolds whose metric tensors have limited regularity. We show that any bi- Lipschitz conformal mapping or 1-quasiregular mapping between two manifolds with Cr metric tensors (r>1) is a Cr+1 conformal (local) diffeomorphism. This result is due to Iwaniec [12], but we give a new proof of this fact. The proof is based on n-harmonic coor- dinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a p-harmonic coordinate system for 1 <p<∞ on any Riemannian manifold. 1. Introduction This paper addresses the following question: given a conformal mapping between two smooth (= C∞) manifolds having metric tensors of limited regularity, how regular is the mapping? This question is very classical if one considers isometries instead of conformal mappings [4,9,24]: it is known that a distance preserving homeomorphism between two Riemannian manifolds with Cr (r>0) metric tensors is in fact a Cr+1 isometry. We refer to [31] for a recent treatment of this topic and related questions based on the systematic use of harmonic coordinates. There has been considerable interest in the regularity of conformal mappings be- tween domains in Euclidean space. Classical proofs of the Liouville theorem show that any C3 conformal mapping between domains in Rn, n ≥ 3, is a restriction of aM¨obius transformation and therefore C∞.
    [Show full text]
  • Book of Abstracts, P
    St. Petersburg Department of Steklov Mathematical Institute RAS Steklov Mathematical Institute RAS Euler International Mathematical Institute Topology, Geometry, and Dynamics: Rokhlin { 100 Conference dedicated to the 100th anniversary of Vladimir Abramovich Rokhlin Archives of the Mathematisches Forschungsinstitut Oberwolfach August 19 { 23, 2019 PDMI & EIMI The conference is supported by: Simons Foundation Chebyshev Lab. Modern Algebra and Applications Lab. Russian Foundation for Basic Research Contents PARTICIPANTS 5 ABSTRACTS 15 Semen Abramyan. Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes 15 Andrei Alpeev. Decay of mutual information for unique Gibbs measures on trees 16 Mikhail Anikushin. Convergence and Strange Nonchaotic Attractors in Almost Periodic Systems 16 Anton Ayzenberg. Orbit spaces of equivariantly formal torus actions 17 Malkhaz Bakuradze. All extensions of C2 by C2n ¢ C2n are good for the Morava K-theory 17 Polina Baron. The Measure of Maximal Entropy for Minimal Interval Exchange Transformations with Flips on 4 Intervals 18 Yury Belousov. A new algorithm of obtaining semimeander diagrams for knots 19 Mohan Bhupal. Open book decompositions of links of minimally elliptic singularities 21 Pavel Bibikov. Theory of differential invariants in algebraic classification problems 21 Michael Blank. Topological properties of measurable semigroup actions 22 Shimon Brooks. Quantum chaos and rotations of the sphere 23 Victor Buchstaber. Vladimir Abramovich Rokhlin and algebraic topology 23 Dmitri Burago. Examples of exponentially many collisions in a hard ball system 24 Sergey Burian. Dynamics of mechanisms near singular points 25 Rezki Chemlal. Measurable factors of one dimensional cellular automata 27 Vladimir Chernov. Causality and Legendrian linking in higher dimensional and causally simple spacetimes 29 Hichem Chtioui.
    [Show full text]
  • Developments in Mathematics
    Developments in Mathematics VOLUME 26 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California For further volumes: http://www.springer.com/series/5834 Vladimir Gutlyanskii • Vladimir Ryazanov Uri Srebro • Eduard Yakubov The Beltrami Equation A Geometric Approach 123 Vladimir Gutlyanskii Vladimir Ryazanov Department of the Partial Department of the Function Theory Differential Equations Institute of Applied Mathematics Institute of Applied Mathematics and Mechanics and Mechanics National Academy of Sciences of Ukraine National Academy of Sciences of Ukraine Donetsk, Ukraine Donetsk, Ukraine Uri Srebro Eduard Yakubov Faculty of Mathematics Faculty of Sciences Technion-Israel Institute of Technology H.I.T. – Holon Institute of Technology Haifa, Israel Holon, Israel ISSN 1389-2177 ISBN 978-1-4614-3190-9 e-ISBN 978-1-4614-3191-6 DOI 10.1007/978-1-4614-3191-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2012934697 Mathematics Subject Classification (2010): 30C62, 30H35, 34M03, 35A01, 35J46, 35J56, 35J67, 35J70, 35Q15, 35Q35, 37F30, 57M10 c Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]
  • That Math Portraits of Mathematicians As Young Readers
    Real Sociedad Matemática Española All that Math Portraits of mathematicians as young readers Editors Antonio Córdoba José L. Fernández Pablo Fernández Portraits of mathematicians as young readers Contributors Antonio Ambrosetti, Kari Astala, Luis Barreira, Pilar Bayer, Christopher J. Bishop, Aline Bonami, Joaquim Bruna, Luis A. Caffarelli, Fernando Chamizo, Sun-Yung Alice Chang, Antonio Córdoba, David Drasin, Charles Fefferman, José L. Fernández, Pedro Luis García, Evarist Giné, Tadeusz Iwaniec, Joseph J. Kohn, Pierre-Gilles Lemarié-Rieusset, Rafael de la Llave, Gaven Martin, Pertti Mattila, Yves Meyer, David Nualart, Antonio Ros, Stephen Semmes, Luis Á. Seco, Elias M. Stein, Gunther Ulhmann, Alexander Volberg, Stephen Wainger, All that Math Jang-Mei Wu, Paul Yang, Marc Yor. Celebrating the Centennial of Real Sociedad Matemática Española ISBN 978-84- 615-2900-1 9 7 8 8 4 6 1 5 2 9 0 0 1 Revista Matemática Iberoamericana All that Math Portraits of mathematicians as young readers Revista Matem´atica Iberoamericana All that Math Portraits of mathematicians as young readers Celebrating the Centennial of Real Sociedad Matem´atica Espa˜nola Antonio C´ordoba, Jos´eL.Fern´andez, Pablo Fern´andez (Editors) Reservados todos los derechos. Prohibida la reproducci´on total o parcial del contenido de este libro, incluso para uso privado, sin permiso escrito de los editores. c 2011 Revista Matem´atica Iberoamericana ISBN: 978-84-615-2900-1 Dise˜no de la edici´on:PFG. Dise˜no de cubierta: Pepe Herrera Taller Gr´afico. http://www.tallergraficopepeherrera.com. Ilustraci´on de la cubierta. “La quinta luz”, Santiago de la Quintana, 2011. Impresi´on:AventuraGr´afica.
    [Show full text]
  • Nonlinear Analysis and Geometric Function Theory
    Abstract and Applied Analysis Nonlinear Analysis and Geometric Function Theory Guest Editors: Ljubomir B. C ´iric´ , Samuel Krushkal, Qamrul Hasan Ansari, David Kalaj, and Vesna Manojlovic´ Nonlinear Analysis and Geometric Function Theory Abstract and Applied Analysis Nonlinear Analysis and Geometric Function Theory Guest Editors: Ljubomir B. Ciri´ c,´ Samuel Krushkal, Qamrul Hasan Ansari, David Kalaj, and Vesna Manojlovic´ Copyright © 2014 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board Ravi P. Agarwal, USA Jaeyoung Chung, Republic of Korea Lorenzo Giacomelli, Italy Bashir Ahmad, Saudi Arabia Silvia Cingolani, Italy Jaume Gine,´ Spain M. O. Ahmedou, Germany Jean M. Combes, France Valery Y. Glizer, Israel Nicholas D. Alikakos, Greece Monica Conti, Italy Laurent Gosse, Italy Debora Amadori, Italy Diego Cordoba,´ Spain Jean P. Gossez, Belgium Pablo Amster, Argentina Juan C. Cortes,´ Spain Jose L. Gracia, Spain Douglas R. Anderson, USA Graziano Crasta, Italy Maurizio Grasselli, Italy Jan Andres, Czech Republic Bernard Dacorogna, Switzerland Qian Guo, China Giovanni Anello, Italy Vladimir Danilov, Russia Yuxia Guo, China Stanislav Antontsev, Portugal Mohammad T. Darvishi, Iran Chaitan P. Gupta, USA Mohamed Kamal Aouf, Egypt Luis F. Pinheiro de Castro, Portugal Uno Hamarik,¨ Estonia Narcisa C. Apreutesei, Romania T. Diagana, USA Ferenc Hartung, Hungary Natig M. Atakishiyev, Mexico Jesus´ I. D´ıaz, Spain Behnam Hashemi, Iran Ferhan M. Atici, USA Josef Dibl´ık, Czech Republic Norimichi Hirano, Japan Ivan G.
    [Show full text]
  • Distortion of Hausdorff Measures and Improved Painlevé Removability For
    DISTORTION OF HAUSDORFF MEASURES AND IMPROVED PAINLEVE¶ REMOVABILITY FOR QUASIREGULAR MAPPINGS K. ASTALA, A. CLOP, J. MATEU, J. OROBITG AND I. URIARTE-TUERO Abstract. The classical Painlev¶e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general K-quasiregular mappings 2 in planar domains the corresponding critical dimension is K+1 . We show that when K > 1, unexpectedly one has improved removabil- 2 ity. More precisely, we prove that sets E of σ-¯nite Hausdor® K+1 - measure are removable for bounded K-quasiregular mappings. On 2 the other hand, dim(E) = K+1 is not enough to guarantee this prop- erty. We also study absolute continuity properties of pull-backs of Hausdor® measures under K-quasiconformal mappings, in particular 2 at the relevant dimensions 1 and K+1 . For general Hausdor® mea- sures Ht, 0 < t < 2 , we reduce the absolute continuity properties to an open question on conformal mappings, see Conjecture 2.3. 1. Introduction A homeomorphism Á : ­ ! ­0 between planar domains ­; ­0 ½ C is 1;2 called K-quasiconformal if it belongs to the Sobolev space Wloc (­) and satis¯es the distortion inequality max j@®Áj · K min j@®Áj a.e. in ­ (1.1) ® ® It has been known since the work of Ahlfors [3] that quasiconformal map- pings preserve sets of zero Lebesgue measure. It is also well known that they preserve sets of zero Hausdor® dimension, since K-quasiconformal mappings are HÄoldercontinuous with exponent 1=K, see [20]. However, these maps do not preserve Hausdor® dimension in general, and it was in the work 2000 Mathematics Subject Classi¯cation.
    [Show full text]