Notions of Convexity pp 225-314 | Cite as

# Plurisubharmonic Functions

## Abstract

In addition to the definition of plurisubharmonic functions Section 4.1 presents the basic facts concerning limits and mean value properties. A new feature compared to the parallel Section 3.2 is that there is a class of associated pseudo-convex sets which have the same relation to plurisubharmonic functions as convex sets have to convex functions. In Section 4.2 existence theorems for the Cauchy-Riemann equations in several complex variables are proved in such sets for *L*^{2} spaces with respect to weights *e*^{−}*ϕ* where ϕ is plurisubharmonic. This gives the tools required in Section 4.3 to study the Lelong numbers of plurisubharmonic functions, describing the dominating associated mass distributions. The study of Lelong numbers is extended to closed positive currents in Section 4.4. A brief discussion of exceptional sets is given in Section 4.5; we just show that the natural exceptional sets are defined by local conditions and closed under countable unions. Instead we pass in Section 4.6 to the study of subclasses of pseudo-convex sets: (weakly) linearly convex sets and C convex sets. These are modelled on the definition of convex sets by supporting planes and intersections with lines, respectively. In section 4.7 we discuss analytic functionals and their Laplace transforms. Besides an analogue of the Paley-Wiener theorem we prove that more refined support properties related to C convex sets can be detected from properties of the Laplace transforms, or rather the Fantappiè transforms obtained from them by a generalization of the classical Borel transform.

## Keywords

Compact Subset Power Series Expansion Complex Line Subharmonic Function Plurisubharmonic Function## Preview

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