## Pluripotential TheoryPluripotential 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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### Contents

Complex differentiation | 3 |

Subharmonic and plurisubharmonic functions | 20 |

The complex MongeAmpere operator | 87 |

5 other sections not shown

### Common terms and phrases

ball Bedford and Taylor biholomorphic C-linear Cegrell classical Green function Clearly compact open subset compact set compact set K C compact subset complex variable Consequently constant converges convex functions Corollary ddcu ddcu)n ddcv decreasing sequence Define Demailly denote Dirichlet problem domain in Cn E C Cn estimate exists formula func harmonic functions hence holomorphic functions holomorphic mapping hyperconvex implies inequality Klimek L-regular Lebesgue measure Lemma Let ft limsup locally bounded locally uniformly main approximation theorem matrix maximal maximum principle Monge-Ampere equation Monge-Ampere operator Moreover neighbourhood non-negative Note o/Cn open set open subset pluricomplex Green function pluriharmonic pluripolar sets plurisubharmonic functions Proof Let properties Proposition prove pseudoconvex relatively compact open right-hand side Sadullaev 1981 satisfies set in Cn Siciak subharmonic functions submanifold subset of Cn Suppose Take topology upper semicontinuous VSH(Sl