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

Research output: ThesisDoctoral Thesis (monograph)

## 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.
Original language English Doctor Mathematics (Faculty of Engineering) Mikael, Passare, Supervisor, External person 1992 Dec 10 91-7153-078-9 Published - 1992 Yes

### Bibliographical note

Defence details

Date: 1992-12-10
Time: 10:00

External reviewer(s)

Name: Ronkin, Lev Isaakovich
Title: Professor
Affiliation: the Engineering Physics Institute for Low Temperatures of the National Academy of Sciences of Ukraine

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• Mathematics

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