Advanced Courses in Mathematics CRM Barcelona

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2 Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta More information about this series at

3 Giovanna Citti Loukas Grafakos Carlos Pérez Alessandro Sarti Xiao Zhong Harmonic and Geometric Analysis Editor for this volume: Joan Mateu (Universitat Autònoma de Barcelona)

4 Giovanna Citti Dipartimento di Matematica Universit à di Bologna Bologna, Italy Carlos Pérez Departamento de Análisis Matemático Universidad de Sevilla Sevilla, Spain Xiao Zhong Department of Mathematics and Statistics University of Jyväskylä Jyväskylä, Finland Loukas Grafakos Department of Mathematics University of Missouri Columbia, MO, USA Alessandro Sarti Dipartimento di Elettronica, Informatica e Sistemistica Universit à di Bologna Bologna, Italy ISSN ISSN (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): Primary: 42B20, 42B37; Secondary: 35R03 Springer Basel Heidelberg New York Dordrecht London Springer Basel 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (

5 Contents Foreword ix 1 Models of the Visual Cortex in Lie Groups Giovanna Citti and Alessandro Sarti 1.1 Introduction Perceptual completion phenomena Gestalt rules and association fields The phenomenological model of elastica Functional architecture of the visual cortex The retinotopic structure The hypercolumnar structure The neural circuitry The visual cortex modeled as a Lie group A first model in the Heisenberg group A sub-riemannian model in the rototranslation group Hörmander vector fields and sub-riemannian structures Connectivity property Control distance Riemannian approximation of the metric Geodesics and elastica Activity propagation and differential operators in Lie groups Integral curves, association fields, and the experiment of Bosking Differential calculus in a sub-riemannian setting Sub-Riemannian differential operators Regular surfaces in a sub-riemannian setting v

6 vi Contents Maximum selectivity and lifting images to regular surfaces Definition of a regular surface Implicit function theorem Non-regular and non-linear vector fields Completion and minimal surfaces A completion process Minimal surfaces in the Heisenberg group Uniform regularity for the Riemannian approximating minimal graph Regularity of the viscosity minimal surface Foliation of minimal surfaces and completion result Bibliography Multilinear Calderón Zygmund Singular Integrals Loukas Grafakos 2.1 Introduction Bilinear Calderón Zygmund operators Endpoint estimates and interpolation for bilinear Calderón Zygmund operators The bilinear T 1 theorem Orthogonality properties for bilinear multiplier operators The bilinear Hilbert transform and the method of rotations Counterexample for the higher-dimensional bilinear ball multiplier 83 Bibliography Singular Integrals and Weights Carlos Pérez Summary Introduction Three applications of the Besicovitch covering lemma to the maximal function Two applications of Rubio de Francia s algorithm: Optimal factorization and extrapolation The sharp factorization theorem The sharp extrapolation theorem Three more applications of Rubio de Francia s algorithm Building A 1 weights from duality

7 Contents vii Improving inequalities with A weights The sharp reverse Hölder property of A 1 weights Main lemma and proof of the linear growth theorem Proof of the logarithmic growth theorem Properties of A p weights Improvements in terms of mixed A 1 -A constants Quadratic estimates for commutators A preliminary result: a sharp connection between the John Nirenberg theorem and the A 2 class Results in the A p context Examples The A 1 case Rearrangement type estimates Proof of Theorem The exponential decay lemma Bibliography De Giorgi Nash Moser Theory Xiao Zhong 4.1 Introduction Equations Motivation: a variational problem Sobolev spaces A brief introduction to Sobolev spaces Definition of weak solutions Moser s iteration Harnack s inequality Weak Harnack s inequality: sup Weak Harnack s inequality: inf De Giorgi s method De Giorgi s class of functions Boundedness of functions in DG(Ω,γ) Hölder continuity of functions in DG(Ω,γ) Further discussions Degenerate elliptic equations Bibliography

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9 Foreword From February to July 2009, the CRM organised a research programme entitled Harmonic Analysis, Geometric Measure Theory and Quasiconformal Mappings. As part of the programme, several advanced courses were delivered by leading experts in the respective fields. This volume contains expositions of the material covered during some of the courses. As the title suggests, the aim of the courses was to deal with selected topics in harmonic analysis and partial differential equations. The first chapter of these notes, written by G. Citti and A. Sarti, explains how the Heisenberg group can be used to model human vision; regularity results for some PDE naturally arising are thoroughly discussed. In the second chapter, L. Grafakos presents some aspects of the basic theory of multilinear harmonic analysis and explains recent developments of the subject. In the third chapter, C. Pérez offers a short introduction to some aspects of the theory of singular integral operators, including a number of new results on sharp weighted bounds for Calderón Zygmund type operators. In the last chapter, X. Zhong presents the De Giorgi Nash Moser theory on regularity of second-order, linear, elliptic equations in divergence form. Thanks are due to the Centre de Recerca Matemàtica for organising and sponsoring the research programme, and to the Centre s administrative staff for smoothly working out innumerable details. Finally, we are grateful to all the participants for their interest in the event and for their positive response. Joan Mateu ix