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

(1992) In Department of Mathematics, University of Stockholm
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
• Professor Ronkin, Lev Isaakovich, the Engineering Physics Institute for Low Temperatures of the National Academy of Sciences of Ukraine
publishing date
type
Thesis
publication status
published
subject
in
Department of Mathematics, University of Stockholm
defense location
Fysikum, Vanadisvägen 9, Stockholm
defense date
1992-12-10 10:00
ISBN
91-7153-078-9
language
English
LU publication?
no
id
e3a8460c-d0fa-41b5-800b-43b6b1b7b49d (old id 1465084)
date added to LUP
2011-02-17 16:41:17
date last changed
2018-11-21 21:16:42
@phdthesis{e3a8460c-d0fa-41b5-800b-43b6b1b7b49d,
abstract     = {The classical Levin-Pfluger theory of entire functions of completely regular growth ($CRG$) of finite<br/><br>
order $\rho$ in one variable establishes a relation between the distribution of zeros of an entire<br/><br>
function and its growth. The most important and interesting result in this theory is the fundamental<br/><br>
principle for $CRG$ functions. In the book of Gruman and Lelong, this basic theorem was<br/><br>
generalized to entire functions of several variables. In this theorem the additional hypotheses<br/><br>
have to be made for integral order $\rho$. We prove one common characterization for<br/><br>
any $\rho$. As an application we prove the following fact: $r^{-\rho} \log<br/><br> |f(rz)|$ converges to the indicator function $h^\ast_f(z)$ as a distribution if and only if $r^{-\rho}<br/><br> \Delta\log |f(rz)|$ converges to $\Delta h^\ast_f(z)$ as a distribution. This also strengthens<br/><br>
a result of Azarin. Lelong has shown that the<br/><br>
indicator $h^\ast_f$ is no longer continuous in several variables. But<br/><br>
Gruman and Berndtsson have proved that $h^\ast_f$ is continuous if the density of<br/><br>
the zero set of $f$ is very small. We relax their conditions. We also get a<br/><br>
characterization of regular growth functions with continuous indicators. Moreover,<br/><br>
we characterize several kinds of limit sets in the sense of Azarin.<br/><br>
<br/><br>
For subharmonic $CRG$ functions in a cone, the situation is much different from functions defined in the<br/><br>
whole space. We introduce a new<br/><br>
definition for $CRG$ functions in a cone. We also give new criteria for<br/><br>
functions to be $CRG$ in an open cone, and strengthen some results due to Ronkin.<br/><br>
Furthermore, we study $CRG$ functions in a closed cone.<br/><br>
<br/><br>
It was proved by Bedford and Taylor that the complex Monge-Amp\ere operator<br/><br>
$(dd^c)^q$ is continuous under monotone limits. Cegrell and Lelong showed<br/><br>
that the monotonicity hypothesis is essential. Improving a result of Ronkin, we get that $(dd^c)^q$ is<br/><br>
continuous under almost uniform limits with respect to Hausdorff $\alpha$-content.<br/><br>
Moreover, we study the Dirichlet problem for the<br/><br>
complex Monge-Amp\ere operator.<br/><br>
<br/><br>
Finally, we confirm a conjecture of Bloom on a generalization of the<br/><br>
M\"untz-Sz\'asz theorem to several variables.},
author       = {Xing, Yang},
isbn         = {91-7153-078-9},
language     = {eng},
series       = {Department of Mathematics, University of Stockholm},
title        = {Zeros and growth of entire functions of several variables, the complex Monge-Ampere operator and some related topics},
year         = {1992},
}