Partial Differential Equations in Several Complex VariablesThis book is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. In the last few decades, significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables. After the background material in complex analysis is developed in Chapters 1 to 3, thenext three chapters are devoted to the solvability and regularity of the Cauchy-Riemann equations using Hilbert space techniques. The authors provide a systematic study of the Cauchy-Riemann equations and the \bar\partial-Neumann problem, including Hórmander's L2 existence progress on the globalregularity and irregularity of the \bar\partial-Neumann operators. The second part of the book gives a comprehensive study of the tangential Cauchy-Riemann equations, another important class of equations in several complex variables first studied by Lewy. An up-to-date account of the L2 theory for \bar\partial b operator is given. Explicit integral solution representations are constructed both on the Heisenberg groups and on strictly convex boundaries with estimates in Hölder and L2spaces. Embeddability of abstract CR structures is discussed in detail here for the first time.Titles in this series are co-published with International Press, Cambridge, MA. |
Contents
| 1 | |
THE CAUCHY INTEGRAL FORMULA AND ITS APPLICATIONS | 15 |
HOLOMORPHIC EXTENSION AND PSEUDOCONVEXITY | 35 |
L2 THEORY FOR d ON PSEUDOCONVEX DOMAINS | 59 |
THE 9NEUMANN PROBLEM ON STRONGLY PSEUDOCONVEX MANIFOLDS | 87 |
BOUNDARY REGULARITY FOR d ON PSEUDOCONVEX DOMAINS | 121 |
CAUCHYRIEMANN MANIFOLDS AND THE TANGENTIAL CAUCHYRIEMANN COMPLEX | 165 |
SUBELLIPTIC ESTIMATES FOR SECOND ORDER DIFFERENTIAL EQUATIONS AND Db | 177 |
THE TANGENTIAL CAUCHYRIEMANN COMPLEX ON PSEUDOCONVEX CR MANIFOLDS | 207 |
FUNDAMENTAL SOLUTIONS FOR Db ON THE HEISENBERG GROUP | 235 |
INTEGRAL REPRESENTATIONS FOR 0 AND db | 261 |
EMBEDDABILITY OF ABSTRACT CR STRUCTURES | 315 |
Other editions - View all
Partial Differential Equations in Several Complex Variables So-chin Chen,Mei-Chi Shaw No preview available - 2001 |
Partial Differential Equations in Several Complex Variables So-Chin Chen,Mei-Chi Shaw No preview available - 2001 |
Common terms and phrases
assume Bergman kernel Bergman projection bounded domain bounded pseudoconvex domain C2 boundary Cauchy Cauchy-Riemann equation Chapter coefficients compact support complex manifold condition coordinates Corollary CR function CR manifold CR structure defining function definition denote dense differential operator distribution sense Dom(a domain in Cn domain of holomorphy equation exists a constant Əzi Əzk follows ƒ ² global Heisenberg group Hence Hermitian Hölder holomorphic function hypersurface implies independent L²(D Lemma Levi form linear Neumann problem norm O-Neumann operator obtain open neighborhood partition of unity proof of Theorem Proposition proves the theorem pseudoconvex CR manifold q)-forms real analytic satisfies smooth bounded pseudoconvex Sobolev spaces solvability strictly plurisubharmonic strongly pseudoconvex CR strongly pseudoconvex domain subelliptic sufficiently small Szegö projection tangential Cauchy-Riemann variables vector fields
Popular passages
Page 5 - The definition is easily seen to be independent of the choice of the local coordinate systems.
Page 8 - Let M be a complex manifold of complex dimension n. and let (¿i,...
Page xii - Committee members assisted us throughout the organization of the conference, and we would like to thank them all. We would also like to thank Microsoft for providing the software used in the Program Committee deliberations.
Page 7 - The manifold E is called the total space, and M is called the base space. The vector space F is called the standard fiber, and its dimension (over the field of scalars K) is called the rank of the bundle.