Pluripotential theory is a recently developed non-linear complex counterpart of classical potential theory. Its main area of application is multidimensional complex analysis. The central part of the pluripotential theory is occupied by maximal plurisubharmonic functions and the generalized complex Monge-Ampere operator. The interplay between these two concepts provides the focal point of this monograph, which contains an up-to-date account of the developments from the large volume of recent work in this area. A substantial proportion of the work is devoted to classical properties of subharmonic and plurisubharmonic functions, which makes the pluripotential theory available for the first time to a wide audience of analysts.
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Subharmonic and plurisubharmonic functions
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a e Q Bedford and Taylor C-linear Cegrell Co(Q compact set compact subset Consequently continuous converges convex convex functions Corollary ddºu ddºv decreasing sequence Define Demailly denote Dirichlet problem domain erists formula function for Q function u e go(z harmonic functions hence holomorphic functions holomorphic mapping hyperconver implies Klimek L-regular Lebesgue measure Lemma Let Q Let u e lim sup locally bounded locally uniformly maximum principle measure Monge–Ampère equation Monge–Ampère operator Moreover neighbourhood non-negative open set open subset pluricomplex Green function pluriharmonic pluripolar sets plurisubharmonic functions point a e polynomial positive Proof Let properties Proposition prove pseudoconvex PSH(Q Q C C result satisfies Shi Q Siciak ſº subharmonic functions subset of Q topology u e C*(Q u e PSH u e PSH Q upper semicontinuous z e Q