64 research outputs found

    Subextension of plurisubharmonic functions with weak singularities

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    We prove several results showing that plurisubharmonic functions with various bounds on their Monge-Ampere masses on a bounded hyperconvex domain always admit global plurisubharmonic subextension with logarithmic growth at infinity

    Maximal subextensions of plurisubharmonic functions

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    In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Amp\`ere measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Amp\`ere measure and the right bound on its Monge-Amp\`ere mass on the unit ball in \C^n for which the maximal subextension to the complex projective space \mb P_n does not have a globally well defined complex Monge-Amp\`ere measure

    Weighted pluricomplex energy

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    We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy Eχ(Ω)\mathcal{E}_\chi (\Omega) in the general case (χ(0)=0\chi(0)=0 i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator (ddc)n(dd^c \cdot)^n on the classes Eχ(Ω).\mathcal{E}\chi(\Omega).Comment: Contrary to what we claimed in the previous version, in Theorem 5.1 we generalize some Theorem of Urban Cegrell but we do not give a new proof. To appear in Potenial Analysi

    Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

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    We give a sufficient condition on the Monge-Amp\`ere mass of a plurisubharmonic function uu for exp(2u)\exp (- 2 u) to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.Comment: extended version with new results and more application

    Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

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    First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page
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