Pseudoconvex classes of functions. I: Pseudoconcave and pseudoconvex sets
This is a joint review for parts I-III. Natural classes of subharmonic, plurisubharmonic, convex and q- plurisubharmonic functions have a number of common properties which can be reduced to several basic ones. Taking these as axioms, the author defines in Part I the general notion of a pseudoconvex class of functions. The main axioms deal with localization properties and preservation of the class by operations like taking limits of decreasing sequences and maximum of several functions. One of the difficulties is that pseudoconvex classes are usually not closed with respect to addition; however, the set of function whose addition preserves the class is assumed to separate the points of the base space. As an example of a natural result obtained in this context we mention the existence and uniqueness theorem for the Bremermann-Dirichlet problem, with solutions in a pseudoconvex class P. A new and crucial tool used here (and throughout the papers) is the notion of the dual pseudoconvex class \(P^ d\) which consists of functions f such that \(f+g\) satisfies the local maximum principle for all \(g\in P.\) Aside for duality, Part I introduces P-pseudoconvex set and P-maximum sets. The latter are relatively closed sets on which functions of class P satisfy the local maximum principle. Then P-pseudoconvex set (which are analogs of the usual pseudoconvex sets) are defined by a form of Kontinuitätssatz, with P-maximum sets replacing analytic discs. One of the motivations of this study of pseudoconvex classes comes from the applications to complex interpolation theory given by the author [Trans. Am. Math. Soc. 308, No.2, 685-711 (1988; Zbl 0657.46057)]. The connection is based on the fact that the normed spaces \(\{({\mathbb{C}}^ n,\| \cdot \|_ z):\) \(z\in G\}\), \(G\subset {\mathbb{C}}^ k\), form an interpolation family relative to a given pseudoconvex class P if and only if the set \(\{(z,w)\in G\times {\mathbb{C}}^ n:\) \(\| w\|_ z<1\}\) is P-pseudoconvex, and its closure in \(G\times {\mathbb{C}}^ n\) is a P-maximum set. The structure of a pseudoconvex class can be fairly well described when the class is invariant with respect to the group of translations of \(R^ n\) (Part II) or, more generally, with respect to a real Lie group acting transitively on a homogeneous space (Part III). The regularization method introduced by the author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 303-326 (1984; Zbl 0583.32046)] allows to show that every invariant pseudoconvex class has a dense subclass consisting of functions having almost everywhere second-order derivatives, and so it can be characterized in terms of the Hessian matrix. In particular, translation- invariant pseudoconvex classes on \(R^ n\) are in one-to-one correspondence with closed (convex) cones of symmetric \(n\times n\) matrices that are preserved by addition of positive-definite matrices. This is used to show that functions of a pseudoconvex class can be approximated by piecewise smooth functions of the same class. (In case of q-plurisubharmonic functions this result was independently obtained by \textit{L. Bungart} [Piecewise smooth approximationss to q-plurisubharmonic functions, Pac. J. Math. (to appear)]. This result is nearly optimal, as smooth approximation is generally not possible. Many new examples of pseudoconvex classes are given in Parts II and III. In the latter, pseudoconvex classes related to the natural bundle structure on \(G\times {\mathbb{C}}^ n\) are studied. The main result characterizes corresponding dual classes and is critical in the proof of duality for complex interpolation method in the above first cited paper of the author.
scientific article; zbMATH DE number 4134423