We study the complex Monge-Ampère operator on the classes of finite pluricomplex energy E χ ( ) in the general case (χ(0) = 0 i.e. the total MongeAmpère 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 (dd c ·) n on the classes Eχ( ).
Abstract. First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean bail B C C ~ with its relative logarithmic capacity in C ~ with respect to the same ball B. An analogous 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 plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity on C ~ as well as the Cegrell class of plurisubharmonic functions of bounded Monge Ampere mass on a hypereonvex domain f~ 9 ~ .Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonie functions.
Let X be a compact Kähler manifold and ω be a smooth closed form of bidegree (1, 1) which is nonnegative and big. We study the classes E χ (X, ω) of ω-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight χ has fast growth at infinity, the corresponding functions are close to be bounded.We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class E χ (X, ω). This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions.Our result extends those of U. Cegrell's and S. Kolodziej's and puts them into a unifying frame. It also gives a simple proof of S. T. Yau's celebrated a priori C 0 -estimate.
In this paper we give some properties of the weighted energy class εχ and study the approximation in these classes. We prove that any function in εχ can be approximated by an increasing sequence of plurisubharmonic functions defined on larger domains and with finite χ-energy.
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