Zeros and growth of entire functions of several variables, the complex Monge-Ampère operator and some related topics

Abstract: The classical Levin-Pfluger theory of entire functions of completely regular growth (CRG) of finite order p in one variable establishes a relation between the distribution of zeros of an entire function and its growth. The most important and interesting result in this theory is the fundamental principle for CRG functions. In the book of Gruman and Lelong, this basic theorem was generalized to entire functions of several variables. In this theorem the additional hypotheses have to be made for integral order p. We prove one common characterization for any p. As an application we prove the following fact: r_plog |/(rz)| converges to the indicator function hj(z) as a distribution if and only if r-,)Alog |/(rz)| converges to Ahy(z) as a distribution. This also strengthens a result of Azarin. Lelong has shown that the indicator hf is no longer continuous in several variables. But Gruman and Bemdtsson have proved that hy is continuous if the density of the zero set of / is very small. We relax their conditions. We also get a characterization of regular growth functions with continuous indicators. Moreover, we characterize several kinds of limit sets in the sense of Azarin.For subharmonic CRG functions in a cone, the situation is much different from functions defined in the whole space. We introduce a new definition for CRG functions in a cone. We also give new criteria for functions to be CRG in an open cone, and strengthen some results due to Ronkin. Furthermore, we study CRG functions in a closed cone.It was proved by Bedford and Taylor that the complex Monge-Ampere operator (ddc)q is continuous under monotone limits. Cegrell and Lelong showed that the monotonicity hypothesis is essential. Improving a result of Ronkin, we get that {ddc)q is continuous under almost uniform limits with respect to Hausdorff a-content. Moreover, we study the Dirichlet problem for the complex Monge-Ampere operator.Finally, we confirm a conjecture of Bloom on a generalization of the Miintz-Szasz theorem to several variables.