Jensen measures, duality and pluricomplex Green functions
Författare
Summary, in English
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.
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.
Publiceringsår
1999
Språk
Engelska
Dokumenttyp
Doktorsavhandling
Ämne
- Mathematics
Nyckelord
- analytic discs
- hyperconvexity
- Lempert function
- Jensen measures
- pluricomplex Green functions
- boundary values of plurisubharmonic functions
- pluripotential theory
Status
Published
Handledare
- Urban Cegrell
ISBN/ISSN/Övrigt
- ISBN: 91-7191-701-2
Försvarsdatum
12 november 1999
Försvarstid
10:00
Försvarsplats
Umeå university
Opponent
- Finnur Larusson (Prof)