Jensen measures, duality and pluricomplex Green functions
(1999) Abstract
 This thesis conceptually consists of two parts. The fist partthe
first half of paper I and papers IIIVis 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... (More)  This thesis conceptually consists of two parts. The fist partthe
first half of paper I and papers IIIVis 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 Bregular 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 MongeAmpère 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. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1314603
 author
 Wikström, Frank ^{LU}
 supervisor
 opponent

 Prof Larusson, Finnur, University of Western Ontario
 publishing date
 1999
 type
 Thesis
 publication status
 published
 subject
 keywords
 analytic discs, hyperconvexity, Lempert function, Jensen measures, pluricomplex Green functions, boundary values of plurisubharmonic functions, pluripotential theory
 defense location
 Umeå university
 defense date
 19991112 10:00
 ISBN
 9171917012
 language
 English
 LU publication?
 no
 id
 519d7ac8c57140bca61cee022167ceb6 (old id 1314603)
 date added to LUP
 20110217 16:44:24
 date last changed
 20180529 10:05:07
@phdthesis{519d7ac8c57140bca61cee022167ceb6, abstract = {This thesis conceptually consists of two parts. The fist partthe<br/><br> first half of paper I and papers IIIVis a study of Jensen<br/><br> measures and their role in pluripotential theory. Lately, there have<br/><br> been a great interest in new methods for constructing plurisubharmonic<br/><br> functions as lower envelopes of disc functionals in the spirit of<br/><br> Poletsky. In this context, Jensen measures of various types<br/><br> play a significant role.<br/><br> <br/><br> The main results in this part are the following: In paper I, we give a<br/><br> characterisation of hyperconvex domains in terms of Jensen measures<br/><br> for boundary points. This result is applied to give a geometric<br/><br> interpretation of hyperconvex Reinhardt domains. Paper II is a study<br/><br> of different classes of Jensen measures and their relation. In<br/><br> particular, it is shown that Jensen measures for continuous<br/><br> plurisubharmonic functions and Jensen measures for upper bounded<br/><br> plurisubharmonic functions coincide in Bregular domains. This is<br/><br> done through an approximation result of independent interest. Paper II<br/><br> also contains a characterisation of boundary values of<br/><br> plurisubharmonic functions in terms of Jensen measures. Such a<br/><br> characterisation is useful in the study of the Dirichlet problem for<br/><br> the complex MongeAmpère operator. In paper III, we study the<br/><br> geometry of continuous maximal plurisubharmonic functions. It is known<br/><br> that a sufficiently smooth maximal plurisubharmonic function whose<br/><br> complex Hessian is of constant rank induces a foliation such that the<br/><br> function is harmonic along the leaves of the foliation. Using a<br/><br> structure theorem by Duval and Sibony, we show that to every<br/><br> continuous maximal plurisubharmonic function, one can find a family of<br/><br> positive (1,1)currents, such that the function is harmonic along<br/><br> these currents. Paper IV is a study of representing measures and their<br/><br> bounded point evaluations. The main result is an example showing that<br/><br> the set of bounded point evaluations may be a proper subset of the<br/><br> polynomial hull of the support of the measure.<br/><br> <br/><br> The second part of the thesis, the second half of paper~I and papers V<br/><br> and VI, is a study of the pluricomplex Green function and various<br/><br> variations of it. These functions are important in many areas of<br/><br> complex analysis, not only in pluripotential theory.<br/><br> <br/><br> In this second part, the main results are the following: In paper I we study<br/><br> the behaviour of the pluricomplex Green function as the pole tends to<br/><br> the boundary. In particular, we prove that for every bounded<br/><br> hyperconvex domain, there is an exceptional pluripolar set outside of<br/><br> which the upper limit of $g(z,w)$ is zero as $w$ tends to the boundary.<br/><br> This result has recently been used to show that every bounded<br/><br> hyperconvex domain is Bergman complete. Paper I also contains an<br/><br> explicit formula for the pluricomplex Green function in the Hartogs'<br/><br> triangle. Paper V is a study of the set where the multipole Lempert<br/><br> function coincides with the sum of the individual single pole<br/><br> functions. The main result is that in bounded convex domains, this set<br/><br> is the union of all complex geodesics connecting the poles. Finally,<br/><br> paper~VI is a study of extremal discs for the multipole Lempert<br/><br> function. Here, the main result is an intrinsic characterisation of<br/><br> these extremal discs.}, author = {Wikström, Frank}, isbn = {9171917012}, keyword = {analytic discs,hyperconvexity,Lempert function,Jensen measures,pluricomplex Green functions,boundary values of plurisubharmonic functions,pluripotential theory}, language = {eng}, title = {Jensen measures, duality and pluricomplex Green functions}, year = {1999}, }