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

Abstract: The classical Levin-Pfluger theory of entire functions of completely regular growth ($CRG$) of finite order $\rho$ 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 $\rho$. We prove one common characterization for any $\rho$. As an application we prove the following fact: $ r^{-\rho} \log |f(rz)|$ converges to the indicator function $h^\ast_f(z)$ as a distribution if and only if $r^{-\rho} \Delta\log |f(rz)|$ converges to $\Delta h^\ast_f(z)$ as a distribution. This also strengthens a result of Azarin. Lelong has shown that the indicator $h^\ast_f$ is no longer continuous in several variables. But Gruman and Berndtsson have proved that $h^\ast_f$ is continuous if the density of the zero set of $f$ 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-Amp\`ere operator $(dd^c)^q$ is continuous under monotone limits. Cegrell and Lelong showed that the monotonicity hypothesis is essential. Improving a result of Ronkin, we get that $(dd^c)^q$ is continuous under almost uniform limits with respect to Hausdorff $\alpha$-content. Moreover, we study the Dirichlet problem for the complex Monge-Amp\`ere operator. Finally, we confirm a conjecture of Bloom on a generalization of the M\"untz-Sz\'asz theorem to several variables.