| Title | Jensen measures, duality and pluricomplex Green functions |
| Authors | Frank Wikström |
| Year | 1999 |
| Document type | Thesis |
| Language | eng |
| Abstract English | This thesis conceptually consists of two parts. The fist part---the<br> first half of paper I and papers II--IV---is a study of Jensen<br> measures and their role in pluripotential theory. Lately, there have<br> been a great interest in new methods for constructing plurisubharmonic<br> functions as lower envelopes of disc functionals in the spirit of<br> Poletsky. In this context, Jensen measures of various types<br> play a significant role.<br> <br> The main results in this part are the following: In paper I, we give a<br> characterisation of hyperconvex domains in terms of Jensen measures<br> for boundary points. This result is applied to give a geometric<br> interpretation of hyperconvex Reinhardt domains. Paper II is a study<br> of different classes of Jensen measures and their relation. In<br> particular, it is shown that Jensen measures for continuous<br> plurisubharmonic functions and Jensen measures for upper bounded<br> plurisubharmonic functions coincide in B-regular domains. This is<br> done through an approximation result of independent interest. Paper II<br> also contains a characterisation of boundary values of<br> plurisubharmonic functions in terms of Jensen measures. Such a<br> characterisation is useful in the study of the Dirichlet problem for<br> the complex Monge-Ampère operator. In paper III, we study the<br> geometry of continuous maximal plurisubharmonic functions. It is known<br> that a sufficiently smooth maximal plurisubharmonic function whose<br> complex Hessian is of constant rank induces a foliation such that the<br> function is harmonic along the leaves of the foliation. Using a<br> structure theorem by Duval and Sibony, we show that to every<br> continuous maximal plurisubharmonic function, one can find a family of<br> positive (1,1)-currents, such that the function is harmonic along<br> these currents. Paper IV is a study of representing measures and their<br> bounded point evaluations. The main result is an example showing that<br> the set of bounded point evaluations may be a proper subset of the<br> polynomial hull of the support of the measure.<br> <br> The second part of the thesis, the second half of paper~I and papers V<br> and VI, is a study of the pluricomplex Green function and various<br> variations of it. These functions are important in many areas of<br> complex analysis, not only in pluripotential theory.<br> <br> In this second part, the main results are the following: In paper I we study<br> the behaviour of the pluricomplex Green function as the pole tends to<br> the boundary. In particular, we prove that for every bounded<br> hyperconvex domain, there is an exceptional pluripolar set outside of<br> which the upper limit of $g(z,w)$ is zero as $w$ tends to the boundary.<br> This result has recently been used to show that every bounded<br> hyperconvex domain is Bergman complete. Paper I also contains an<br> explicit formula for the pluricomplex Green function in the Hartogs'<br> triangle. Paper V is a study of the set where the multipole Lempert<br> function coincides with the sum of the individual single pole<br> functions. The main result is that in bounded convex domains, this set<br> is the union of all complex geodesics connecting the poles. Finally,<br> paper~VI is a study of extremal discs for the multipole Lempert<br> function. Here, the main result is an intrinsic characterisation of<br> these extremal discs. |
| Keywords | analytic discs, hyperconvexity, Lempert function , Jensen measures, pluricomplex Green functions, boundary values of plurisubharmonic functions, pluripotential theory, |
| ISBN/ISSN/Other | ISBN: 91-7191-701-2 |
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