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.

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