This monograph is a valuable introduction to an active area of contemporary research at the intersection of multidimensional complex analysis, operator theory, and partial differential equations. The central topic is the canonical solution operator for the inhomogeneous Cauchy-Riemann equations. The exposition is relatively self-contained, since the book covers relevant prerequisites from functional analysis, Sobolev spaces, spectral theory, and partial differential equations.
Chapter 1 treats the basics of the Bergman kernel function, including the transformation rule under biholomorphic mappings and examples (the unit ball, Fock space). Chapter 2 discusses the canonical solution operator for \(\overline{\partial}\) from the Hilbert-space point of view and characterizes compactness of the operator for certain weighted spaces of entire functions in one dimension. Chapter 3 turns to higher dimensions and examines compactness and membership in Schatten classes of the restriction of the operator to \((0,1)\)-forms with holomorphic coefficients (in weighted spaces where the monomials form an orthogonal family). Chapter 4 develops the Hilbert-space machinery for proving the existence of the \(\overline{\partial}\)-Neumann operator on a bounded pseudoconvex domain in \(\mathbb{C}^{n}\). Chapter 5 is a technical aside about density, the Friedrichs lemma, and Sobolev spaces. Chapters 6 and 7 introduce the weighted \(\overline{\partial}\)-complex and the twisted \(\overline{\partial}\)-complex. Chapter 8 proves Hörmander’s \(L^{2}\)-estimate and applies the result to show infinite-dimensionality of certain weighted spaces of entire functions. Chapter 9 expounds the necessary spectral theory of unbounded self-adjoint operators. Chapter 10 provides background on Schrödinger operators, Pauli and Dirac operators, and the Witten Laplacian, and Chapter 13 discusses when these operators have compact resolvents. Chapter 11 provides some sufficient conditions for compactness of the \(\overline{\partial}\)-Neumann operator: property (P) for bounded pseudoconvex domains, and blowing up at infinity of the Levi eigenvalues of the weight function \(\varphi\) for the space \(L^{2}(\mathbb{C}^{n}, e^{-\varphi})\). Chapter 12 investigates the connection between compactness of the \(\overline{\partial}\)-Neumann operator and compactness of commutators of the Bergman projection with multiplication operators. Chapter 14 determines the spectrum of the \(\overline{\partial}\)-Laplacian and of the Witten Laplacian for the Fock space. The concluding Chapter 15 provides concrete examples in which the canonical solution operator for \(\overline{\partial}\) fails to be compact.
The book is based on the author’s lectures on the \(\overline{\partial}\)-Neumann operator at the University of Vienna, the Erwin Schrödinger International Institute for Mathematical Physics (ESI) in Vienna, and the Centre International de Rencontres Mathématiques (CIRM) in Luminy. The material has some overlap with a recent monograph by E. J. Straube [Lectures on the \(L^2\)-Sobolev theory of the \(\bar\partial\)-Neumann problem. Zürich: European Mathematical Society (EMS) (2010; Zbl 1247.32003)] but in many ways is complementary.