The classical potential theory in the complex plane \({\mathbb{C}}\cong {\mathbb{R}}^ 2\) has been generalized to \({\mathbb{R}}^ n\) a long time ago and the results are in many cases similar to those which are known in \({\mathbb{C}}\). In the \({\mathbb{R}}^ n\)-theory, a crucial role is played by the subharmonic functions and the Laplace operator (which is linear). In particular, the following four fundamental properties (of a closed subset K of an open set \(\Omega \subset {\mathbb{R}}^ n)\) are equivalent in this case: thinness, polarity, negligeability, and removability, and moreover, the compact sets that have the above properties are exactly those with vanishing Newton capacity. The situation completely changes when we pass to the \({\mathbb{C}}^ n\)-theory. First of all, we now have to replace the subharmonic functions by the plurisubharmonic functions and the Laplace operator by the complex Monge-Ampère operator which is non-linear. Consequently, the two theories differ considerably and, in particular, the previous equivalences do not hold longer in the \({\mathbb{C}}^ n\)-case. In both “real” and “complex” cases, a fundamental notion is that of capacity. In the \({\mathbb{C}}^ n\)-case, it is strictly connected with the complex Monge-Ampère operator whose extensive study was started by the fundamental paper by E. Bedford and B. A. Taylor [Invent. Math. 37, 1-44 (1976; Zbl 0315.31007)] and then continued by the same authors in Acta Math. 149, 1-40 (1982; Zbl 0547.32012).
As has been emphasized by the author, the purpose of this book is to study plurisubharmonic and analytic functions in \({\mathbb{C}}^ n\) using capacity theory. In the first three sections the reader is given an introduction to general capacity theory where capacity is thought of as a non-linear generalization of measures. There are discussed, in particular, conditions for capacitability and outer regularity. The “heart” of the monograph are sections V, VI, VII, and VIII. Here the complex Monge-Ampère operator is systematically studied and then applied to investigate certain plurisubharmonic functions (the reader will find there the discussion of quasicontinuity of plurisubharmonic functions with respect to the Monge-Ampère capacity, Dirichlet problem for strictly pseudoconvex, bounded domains in \({\mathbb{C}}^ n\), comparison theorems for the Monge-Ampère operator, \({\mathbb{C}}^ n\)-polarity of negligeable sets, subextendability of plurisubharmonic functions, Green’s function and extremal plurisubharmonic function). In section IX, the gamma capacities of Ronkin and Favorov are discussed (they have been used in connection with removable singularity sets for analytic functions). Section X is devoted to capacities on the boundary of a bounded domain in \({\mathbb{C}}^ n\) whereas in section XI Szegö kernel technics are applied to study some extension problems for analytic functions. The book is closed by section XII where there is considered a capacity generated by representing measures on the spectrum of the algebra of bounded analytic functions.