Zeros and growth of entire functions of several variables, the complex MongeAmpere operator and some related topics
(1992) In Department of Mathematics, University of Stockholm Abstract
 The classical LevinPfluger 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 LevinPfluger 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 MongeAmp\`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 MongeAmp\`ere operator.
Finally, we confirm a conjecture of Bloom on a generalization of the
M\"untzSz\'asz theorem to several variables. (Less)
Please use this url to cite or link to this publication:
http://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
 in
 Department of Mathematics, University of Stockholm
 defense location
 Fysikum, Vanadisvägen 9, Stockholm
 defense date
 19921210 10:00
 ISBN
 9171530789
 language
 English
 LU publication?
 no
 id
 e3a8460cd0fa41b5800b43b6b1b7b49d (old id 1465084)
 date added to LUP
 20110217 16:41:17
 date last changed
 20181121 21:16:42
@phdthesis{e3a8460cd0fa41b5800b43b6b1b7b49d, abstract = {The classical LevinPfluger 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 MongeAmp\`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 MongeAmp\`ere operator.<br/><br> <br/><br> Finally, we confirm a conjecture of Bloom on a generalization of the<br/><br> M\"untzSz\'asz theorem to several variables.}, author = {Xing, Yang}, isbn = {9171530789}, language = {eng}, series = {Department of Mathematics, University of Stockholm}, title = {Zeros and growth of entire functions of several variables, the complex MongeAmpere operator and some related topics}, year = {1992}, }