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Partial differential equations in several complex variables. (English) Zbl 0963.32001
AMS/IP Studies in Advanced Mathematics. 19. Providence, RI: American Mathematical Society (AMS). Somerville, MA: International Press, xii, 380 p. (2001).
This book is an introduction to several complex variables from the viewpoint of partial differential operators. It focuses essentially on two topics: the first (Chapters 1 to 6) is the Dolbeault complex on pseudoconvex domains of \(\mathbb C^n\), the second (Chapters 7 to 12) the tangential Cauchy-Riemann complex on pseudoconvex CR manifolds of hypersurface type. After presenting some classical results and the \(L^2\) theory for pseudoconvex domains in \(\mathbb C^n\) in the first pages, Chapters 5 and 6 discuss the \(\overline\partial\)-Neumann problem and its applications to the study of the properties of the Bergmann projector and of the boundary regularity of biholomorphic maps. There are also two sections devoted to the interesting example of the worm domains.
The second half of the book begins with a definition of CR manifolds of the hypersurface type, of the tangential Cauchy-Riemann complex \(\overline\partial_b\), and with a discussion of the Hans Lewy equation. Chapters 8 and 9 deal with the subelliptic estimates, solvability and Hodge theory for the \(\overline\partial_b\)-complex on pseudocomplex CR hypersurfaces. The next two chapters deal with the \(\square_b\) operator and its fundamental solutions on the Heisenberg group and the integral representations for \(\overline\partial\) and \(\overline\partial_b\), with applications to the \(L^p\) theory and the study of \(\overline\partial_b\) on domains with boundary of pseudoconvex CR hypersurfaces. A final chapter is dedicated to the problem of embeddability of pseudocomplex abstract CR hypersurfaces. The discussion includes Rossi’s and Nirenberg’s examples and the theorem of Boutet de Monvel for compact pseudoconvex abstract CR hypersurfaces of real dimension greater or equal to \(5\).
The book gives an interesting and fairly exhausting overview of the themes of interest and results obtained in the last decades by the school of complex analysis centered about Joseph J. Kohn. It can be considered the best elementary introduction to the study of his work and the work of his students and collaborators.

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32W50 Other partial differential equations of complex analysis in several variables
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators