Jensen measures, duality and pluricomplex Green functions

Abstract

This thesis conceptually consists of two parts. The fist part---the

first half of paper I and papers II--IV---is a study of Jensen

measures and their role in pluripotential theory. Lately, there have

been a great interest in new methods for constructing plurisubharmonic

functions as lower envelopes of disc functionals in the spirit of

Poletsky. In this context, Jensen measures of various types

play a significant role.



The main results in this part are the following: In paper I, we give a

characterisation of hyperconvex domains in terms of Jensen measures

for boundary points. This result is applied to give a geometric

interpretation of hyperconvex Reinhardt domains. Paper II is a study

of different classes of Jensen measures and their relation. In

particular, it is shown that Jensen measures for continuous

plurisubharmonic functions and Jensen measures for upper bounded

plurisubharmonic functions coincide in B-regular domains. This is

done through an approximation result of independent interest. Paper II

also contains a characterisation of boundary values of

plurisubharmonic functions in terms of Jensen measures. Such a

characterisation is useful in the study of the Dirichlet problem for

the complex Monge-Ampère operator. In paper III, we study the

geometry of continuous maximal plurisubharmonic functions. It is known

that a sufficiently smooth maximal plurisubharmonic function whose

complex Hessian is of constant rank induces a foliation such that the

function is harmonic along the leaves of the foliation. Using a

structure theorem by Duval and Sibony, we show that to every

continuous maximal plurisubharmonic function, one can find a family of

positive (1,1)-currents, such that the function is harmonic along

these currents. Paper IV is a study of representing measures and their

bounded point evaluations. The main result is an example showing that

the set of bounded point evaluations may be a proper subset of the

polynomial hull of the support of the measure.



The second part of the thesis, the second half of paper~I and papers V

and VI, is a study of the pluricomplex Green function and various

variations of it. These functions are important in many areas of

complex analysis, not only in pluripotential theory.



In this second part, the main results are the following: In paper I we study

the behaviour of the pluricomplex Green function as the pole tends to

the boundary. In particular, we prove that for every bounded

hyperconvex domain, there is an exceptional pluripolar set outside of

which the upper limit of $g(z,w)$ is zero as $w$ tends to the boundary.

This result has recently been used to show that every bounded

hyperconvex domain is Bergman complete. Paper I also contains an

explicit formula for the pluricomplex Green function in the Hartogs'

triangle. Paper V is a study of the set where the multipole Lempert

function coincides with the sum of the individual single pole

functions. The main result is that in bounded convex domains, this set

is the union of all complex geodesics connecting the poles. Finally,

paper~VI is a study of extremal discs for the multipole Lempert

function. Here, the main result is an intrinsic characterisation of

these extremal discs.

• author: Wikström, Frank
• supervisor: Urban Cegrell
• opponent: Prof Larusson, Finnur, University of Western Ontario
• publishing date: 1999
• type: Thesis
• publication status: published
• subject: Mathematics
• keywords: analytic discs, hyperconvexity, Lempert function, Jensen measures, pluricomplex Green functions, boundary values of plurisubharmonic functions, pluripotential theory
• defense location: Umeå university
• defense date: 1999-11-12 10:00:00
• ISBN: 91-7191-701-2
• language: English
• LU publication?: no
• id: 519d7ac8-c571-40bc-a61c-ee022167ceb6 (old id 1314603)
• date added to LUP: 2016-04-04 13:52:57
• date last changed: 2020-03-19 11:44:29