REAL ANALYSIS - USTC

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REAL ANALYSISIbookroot October 20, 2007PrincetonLecturesinAnalysisI Fourier ANALYSIS : An IntroductionII Complex AnalysisIII Real ANALYSIS : Measure Theory, Integration, andHilbert SpacesIV Functional ANALYSIS : Introductionto Further Topics in AnalysisPrinceton Lectures in AnalysisIIIREAL ANALYSISMeasure Theory, Integration, andHilbert SpacesElias M. Stein&Rami ShakarchiPRINCETON UNIVERSITY PRESSPRINCETON AND OXFORDCopyright 2005 by Princeton University PressPublished by Princeton University Press, 41 William Street,Princeton, New Jersey 08540In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TWAll Rights ReservedLibrary of Congress Control Number 2004114065ISBN 978-0-691-11386-9British Library Cataloging-in-Publication Data is availableThe publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printedPrinted on acid-free paper.

Chapter 5. Hilbert Spaces: Several Examples 207 1 The Fourier transform on L2 207 2 The Hardy space of the upper half-plane 213 3 Constant coe–cient partial difierential equations 221 3.1 Weak solutions 222 3.2 The main theorem and key estimate 224 4* The Dirichlet principle 229 4.1 Harmonic functions 234

  Transform, Hilbert