We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs' triangle. Finally we study the behaviour of the pluricomplex Green function $g(z,w)$ as the pole $w$ tends to a boundary point.

2. Dieu, Nguyen Quang

et al.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Let $Omega$ be a bounded domain in $Cn$. We study relations between Jensen measures for continuous and upper bounded plurisubharmonic functions on $Omega$, and give some conditions on $Omega$ that imply that these two classes of measures coincide.

3.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Let $H^2_m$ be the Hilbert function space on the unit ball in
$C{m}$ defined by the kernel $k(z,w) = (1-langle z,w
rangle)^{-1}$. For any weak zero set of the multiplier algebra of
$H^2_m$, we study a natural extremal function, $E$. We investigate
the properties of $E$ and show for example that $E$ tends to $0$ at
almost every boundary point. We also give several explicit examples
of the extremal function and compare the behaviour of $E$ to the
behaviour of $delta^*$ and $g$, the corresponding extremal function
for $H^infty$ and the pluricomplex Green function, respectively.

4.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.

Area, mått och integraler2008In: Människor och matematik - läsebok för nyfikna, Göteborg: Nationellt centrum för matematikutbildning (NCM), 2008, p. 390-Chapter in book (Other (popular science, discussion, etc.))

Abstract [sv]

Populärvetenskaplig uppsats om det moderna areabegreppet.

5.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

We look at numerical computations of the pluricomplex Green function $g$ with two poles of equal weight for the bidisk. The results we obtain strongly suggest that Coman's conjecture holds in this setting, that is that $g$ equals the Lempert function. We also prove this in a special case. Furthermore, we show that Coman's conjecture fails in the case of two poles of different weight in the unit ball of $C2$.

6.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Let $V$ be an analytic variety in a domain $Omega subset Cn$ and
let $K relcomp V$ be a closed subset. By studying Jensen measures
for certain classes of plurisubharmonic functions on $V$, we prove
that the relative extremal function $omega_K$ is continuous on $V$
if $Omega$ is hyperconvex and $K$ is regular.

7.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in B-regular domain. This theorem implies that the two classes of Jensen measures coincide in B-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.

We consider the pluricomplex Green function with multiple poles as
introduced by Lelong. We give a partial solution to a question
concerning the set where the multipole Green function coincides with
the sum of the corresponding single pole Green functions.

9.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

If $u$ is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of $u$ has constant rank, it is known that there exists a foliation by complex manifolds, such that $u$ is harmonic along the leaves of the foliation.
In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension $(1,1)$ such that the function is harmonic along the currents.

Let Omega be a B-regular domain in C-n and let V be a locally irreducible analytic variety in Omega. Given a continuous function phi is an element of C((V) over bar boolean AND partial derivative Omega), we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by f and furthermore that u is continuous on (V) over bar.

11.

Åhag, Per

et al.

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Czyz, Rafal

Lodin, Sam

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Wikström, Frank

Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.

Let $mathcal{P}$ be a connected, non-degenerate analytic polyhedron in $mathbb{C}^n$, $ngeq 2$. In this note we characterize those continuous functions which can be extended to a plurisubharmonic function in $mathcal{P}$.