Lectures on the L 2 -Sobolev Theory of the -Neumann Problem. Emil J. Straube

Transcription

1 Lectures on the L 2 -Sobolev Theory of the -Neumann Problem Emil J. Straube June 3, 2009

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3 Preface In the summer and fall of 2005, I gave a series of lectures at the Erwin Schrödinger International Institute of Mathematical Physics in Vienna on the basic L 2 -Sobolev theory of the -Neumann problem on domains in C n. These lectures, aimed at advanced graduate students and young researchers, produced a set of notes of about eighty pages. In the spring semester of 2006, I taught a graduate course at Texas A&M University on the same topic, revising and somewhat expanding the notes in the process. Subsequently, I set out to further revise the notes and to make them more or less self-contained, as far as the -Neumann problem is concerned. The intent remained the same: to provide a thorough introduction to the topic in the title, in the setting of bounded pseudoconvex domains in C n, that leads up to current research. This monograph is the result. Chapter 2 presents the basic L 2 -theory. Chapter 3 discusses the subelliptic estimates on strictly pseudoconvex domains. From the point of view of leading up to current research, chapter 4 on compactness and chapter 5 on regularity in Sobolev spaces are the most important. For a detailed description of the contents of these chapters, along with historical remarks, I refer the reader to the introductory chapter 1. A word about prerequisites. The reader is assumed to have a solid background in basic complex and functional analysis (including the elementary L 2 -Sobolev theory and distributions). Some knowledge in several complex variables is clearly helpful, if only for motivation. Concerning partial differential equations, not much is assumed. The elliptic regularity of the Dirichlet problem for the Laplacian is quoted a few times. On the other hand, the ellipticity results needed for elliptic regularization in chapter 3 are proved from scratch. I have received institutional support for this project from the Erwin Schrödinger International Institute of Mathematical Physics through a Seiii

4 iv PREFACE nior Research Fellowship, from Texas A&M University through a Faculty Development Leave, and from the National Science Foundation through grants DMS and DMS This support is hereby gratefully acknowledged. I am indebted to the students and colleagues (young and not so young) in Vienna and in College Station who attended the lectures, asked pertinent questions, caught errors, and made suggestions for improvements. I thank them all. Likewise, I thank everybody who has communicated to me errors, typographical, grammatical, mathematical, or historical, in various draft versions of the manuscript. I was involved in some of the results presented here. For the most part, these results were obtained through joint work. I would like to take this opportunity to thank my coauthors over the years for sharing their ideas. College Station, Texas April 2009

5 Contents Preface iii 1 Introduction 1 2 The L 2 -theory The basic L 2 -setup Special boundary charts Sobolev spaces of forms Density of nice forms Weighted norms The twisted Kohn-Morrey-Hörmander formula The role of pseudoconvexity Complex Laplacian and -Neumann operator Applications of real potential theory Restriction of forms to subdomains Solvability of in C(0,q) (Ω) Non-plurisubharmonic weights techniques, the Ohsawa-Takegoshi Theorem Strictly pseudoconvex domains Estimates at the ground level Estimates for operators related to N Sobolev estimates, elliptic regulariziation Pseudolocal estimates Singular holomorphic functions Compactness An application to Toeplitz operators v

6 vi CONTENTS 4.2 General facts Estimates in Sobolev norms A sufficient condition for compactness Analysis of property (P q ) Some examples Hartogs domains in C Obstructions to (P q ) and to compactness Locally convexifiable domains A variant of property (P q ) Geometric sufficient conditions Regularity in Sobolev spaces Kohn s weighted theory A linear solution operator continuous in C (Ω) Bergman projections and -Neumann operators Benign derivatives Good vector fields imply Sobolev estimates Domains with symmetries Commutators: strictly pseudoconvex directions Vector fields, defining functions, Hessians A useful one-form Submanifolds and a cohomology class A foliation in the boundary Worm domains A unified approach to global regularity