Zeros and growth of entire functions of several variables, the complex Monge-Ampere operator and some related topics
(1992)- 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:
https://lup.lub.lu.se/record/1465084
- author
- Xing, Yang LU
- supervisor
- opponent
-
- Professor Ronkin, Lev Isaakovich, the Engineering Physics Institute for Low Temperatures of the National Academy of Sciences of Ukraine
- publishing date
- 1992
- type
- Thesis
- publication status
- published
- subject
- defense location
- Fysikum, Vanadisvägen 9, Stockholm
- defense date
- 1992-12-10 10:00:00
- ISBN
- 91-7153-078-9
- language
- English
- LU publication?
- no
- id
- e3a8460c-d0fa-41b5-800b-43b6b1b7b49d (old id 1465084)
- date added to LUP
- 2016-04-04 13:50:49
- date last changed
- 2020-03-19 15:20:44
@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}},
title = {{Zeros and growth of entire functions of several variables, the complex Monge-Ampere operator and some related topics}},
year = {{1992}},
}