ABSTRACT
Two special classes of plurisubharmonic functions are of particular usefulness and interest and will be discussed briefly here. The first and by far the most important is the subclass of strictly pluriharmonic functions. Recall from Theorem K8 that a real-valued function of class C 2 in an open subset D ⊆ ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq1812.tif"/> is plurisubharmonic in D precisely when its Levi form is positive semidefmite at each point of D. Among the plurisubharmonic functions of class C 2 in D one extreme case evidently consists of those functions with Levi forms identically zero; by Theorem K2 these are just the plurisubharmonic functions in D. The complementary extreme case consists of those functions with Levi forms strictly positive definite at each point of D; these are quite naturally called strictly plurisubharmonic functions and play a useful special role in the theory. Of course, it is not really necessary to restrict these considerations to functions of class C 2 if the Levi form is interpreted in the sense of distributions, as in Theorem K15 and its corollaries, and that is perhaps the most natural way in which to introduce the full class of strictly plurisubharmonic functions. But it is in some ways easier formally to introduce this class of functions in the following equivalent but more primitive form.
