| Title | Zeros and growth of entire functions of several variables, the complex Monge-Ampere operator and some related topics |
| Authors | Yang Xing |
| Publication | Department of Mathematics, University of Stockholm |
| Year | 1992 |
| Document type | Thesis |
| Language | eng |
| Abstract English | The classical Levin-Pfluger theory of entire functions of completely regular growth ($CRG$) of finite<br> order $\rho$ in one variable establishes a relation between the distribution of zeros of an entire<br> function and its growth. The most important and interesting result in this theory is the fundamental<br> principle for $CRG$ functions. In the book of Gruman and Lelong, this basic theorem was<br> generalized to entire functions of several variables. In this theorem the additional hypotheses<br> have to be made for integral order $\rho$. We prove one common characterization for<br> any $\rho$. As an application we prove the following fact: $ r^{-\rho} \log<br> |f(rz)|$ converges to the indicator function $h^\ast_f(z)$ as a distribution if and only if $r^{-\rho}<br> \Delta\log |f(rz)|$ converges to $\Delta h^\ast_f(z)$ as a distribution. This also strengthens<br> a result of Azarin. Lelong has shown that the<br> indicator $h^\ast_f$ is no longer continuous in several variables. But<br> Gruman and Berndtsson have proved that $h^\ast_f$ is continuous if the density of<br> the zero set of $f$ is very small. We relax their conditions. We also get a<br> characterization of regular growth functions with continuous indicators. Moreover,<br> we characterize several kinds of limit sets in the sense of Azarin.<br> <br> For subharmonic $CRG$ functions in a cone, the situation is much different from functions defined in the<br> whole space. We introduce a new<br> definition for $CRG$ functions in a cone. We also give new criteria for<br> functions to be $CRG$ in an open cone, and strengthen some results due to Ronkin.<br> Furthermore, we study $CRG$ functions in a closed cone.<br> <br> It was proved by Bedford and Taylor that the complex Monge-Amp\`ere operator<br> $(dd^c)^q$ is continuous under monotone limits. Cegrell and Lelong showed<br> that the monotonicity hypothesis is essential. Improving a result of Ronkin, we get that $(dd^c)^q$ is<br> continuous under almost uniform limits with respect to Hausdorff $\alpha$-content.<br> Moreover, we study the Dirichlet problem for the<br> complex Monge-Amp\`ere operator.<br> <br> Finally, we confirm a conjecture of Bloom on a generalization of the<br> M\"untz-Sz\'asz theorem to several variables. |
| ISBN/ISSN/Other | ISSN: 91-7153-078-9 |
Questions: webmaster
Last update: 2013-04-11
Centre for Mathematical Sciences, Box 118, SE-22100, Lund. Telefon: +46 46-222 00 00 (vx)