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Several Complex Variables and Integral Formulas TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk SSeevveerraall CCoommpplleexx VVaarriiaabblleess aanndd IInntteeggrraall FFoorrmmuullaass KKeennzzoo AAddaacchhii Nagasaki University, Japan World Scientifi c NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. SEVERAL COMPLEX VARIABLES AND INTEGRAL FORMULAS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-270-574-7 ISBN-10 981-270-574-0 Printed in Singapore. LaiFun - Several Complex.pmd 1 12/4/2006, 11:06 AM December19,2006 18:26 WSPC/BookTrimSizefor9inx6in ws-book9x6 To my family Machiko, Hidehiko and Yuko v December19,2006 18:26 WSPC/BookTrimSizefor9inx6in ws-book9x6 TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk December19,2006 18:26 WSPC/BookTrimSizefor9inx6in ws-book9x6 Preface Theaimofthisbookistostudysomeimportantresultsobtainedinthelast 50yearsinthefunctiontheoryofseveralcomplexvariablesthataremainly concerned with the extension of holomorphic functions from submanifolds of pseudoconvex domains and estimates for solutions of the ∂¯ problem in pseudoconvex domains. This book is divided into five chapters. In Chapter 1 we recall the elementary theory of functions of several complexvariables. Weprovethateverydomainofholomorphyisapseudo- convexopenset. Moreover,wegivetheproofoftheHartogstheoremwhich means that a separately analytic function is analytic. In Chapter 2 we deal with L2 estimates for the ∂¯ problem in pseudoconvex domains in Cn due to Ho¨rmander. As an application, we give the affirmative answer for the Levi problem. Moreover, we prove the Ohsawa- Takegoshi extension theorem by following the method of Jarnicki-Pflug. In Chapter 3 we construct integral formulas for differential forms on bounded domains in Cn with smooth boundary, that is, the Bochner- Martinelli formula, the Koppelman formula, the Leray formula and the Koppelman-Lerayformulaarederived. Usingtheintegralformula,weprove Hölder estimates for the ∂¯problem in strictly pseudoconvex domains with smoothboundary. Moreover,we provebounded andcontinuous extensions of holomorphic functions from submanifolds of strictly pseudoconvex domains with smooth boundary which were proved by Henkin in 1972. We also prove Hp and Ck extensions. Finally, we prove Fefferman’s mapping theorem by following the method of Range. In Chapter 4 we discuss the Berndtsson-Andersson formula and the Berndtsson formula. As an application of the Berndtsson-Andersson formula,wegiveLpestimatesforsolutionsofthe∂¯probleminstrictlypseudo- vii December19,2006 18:26 WSPC/BookTrimSizefor9inx6in ws-book9x6 viii Several Complex Variables and Integral Formulas convex domains in Cn with smooth boundary. Using the Berndtsson formula, we give counterexamples of Lp (p > 2) extensions of holomorphic functions. Finally, we introduce an integral formula which was used by Diederich-Mazzilli to prove bounded extensions of holomorphic functions from affine linear submanifolds of a smooth convex domain of finite type. Chapter 5 is devoted to the study of classical fundamental theorems in the function theory of severalcomplex variables some of which are used to prove theorems in the previous chapters. Appendix A is concerned with the compact operator theory in Banach spaces which is used to prove Fefferman’s mapping theorem. In Appendix B we give solutions to the Exercises. IamgratefultoSaburouSaitohwhosuggestedtomethepublicationof this book. I am also grateful to Heinrich GW Begehr who suggested that World Scientific might be interested in publishing this book. I would like to express my sincere gratitude to JojiKajiwara,Professor Emeritus at the Kyushu University, who introduced me to the function theory of several complex variables when I was a student at the Kyushu University, and to Morisuke Hasumi, Professor Emeritus at the Ibaraki University, who introduced me to the theory of function algebras when I was studying at the Ibaraki University. Finally,IwanttoexpressmythankstoMsZhangJi,MsKwongLaiFun and the staff of World Scientific for their help and cooperation. Kenzo Adachi December19,2006 18:26 WSPC/BookTrimSizefor9inx6in ws-book9x6 Contents Preface vii 1. Pseudoconvexity and Plurisubharmonicity 1 1.1 The Hartogs Theorem . . . . . . . . . . . . . . . . . . . . . 1 1.2 Characterizations of Pseudoconvexity. . . . . . . . . . . . . 19 2. The ∂¯Problem in Pseudoconvex Domains 47 2.1 The Weighted L2 Space . . . . . . . . . . . . . . . . . . . . 47 2.2 L2 Estimates in Pseudoconvex Domains . . . . . . . . . . . 53 2.3 The Ohsawa-TakegoshiExtension Theorem . . . . . . . . . 100 3. Integral Formulas for Strictly Pseudoconvex Domains 117 3.1 The Homotopy Formula . . . . . . . . . . . . . . . . . . . . 117 3.2 Ho¨lder Estimates for the ∂¯Problem . . . . . . . . . . . . . 143 3.3 Bounded and Continuous Extensions . . . . . . . . . . . . . 152 3.4 Hp and Ck Extensions . . . . . . . . . . . . . . . . . . . . . 183 3.5 The Bergman Kernel . . . . . . . . . . . . . . . . . . . . . . 196 3.6 Fefferman’s Mapping Theorem . . . . . . . . . . . . . . . . 210 4. Integral Formulas with Weight Factors 245 4.1 The Berndtsson-AnderssonFormula . . . . . . . . . . . . . 245 4.2 Lp Estimates for the ∂¯Problem . . . . . . . . . . . . . . . . 255 4.3 The Berndtsson Formula . . . . . . . . . . . . . . . . . . . . 264 4.4 Counterexamples for Lp (p>2) Extensions . . . . . . . . . 270 4.5 Bounded Extensions by Means of the Berndtsson Formula . 281 ix