Related papers: Some properties of the complex Monge-Ampere operat…
We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral…
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere…
The aim of the paper is to investigate the structure of plurifinely open sets. As an application, we will prove an equality on complex Monge-Amp\`ere measures in plurifinely open sets.
The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in…
We generalize Monge property of real matrices for interval matrices. We define two classes of interval matrices with Monge property - in a strong and in a weak sense. We study fundamental properties of both classes. We show several…
In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of…
The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Amp\`ere operator can be reasonably…
In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order…
We introduce a non local analog to the Monge-Ampere operator and show some of its properties. We prove that a global problem involving this operator has C 1,1 solutions in the full space.
The Composite Operator Method (COM) is formulated, its internals illustrated in detail and some of its most successful applications reported. COM endorses the emergence, in strongly correlated systems (SCS), of composite operators,…
We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex…
We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$,…
We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\`ere energy. They…
We prove several quantitative stability estimates for solutions of complex Monge-Ampere equations when both the cohomology class and the prescribed singularity vary. In a broad sense, our results fit well into the study of degeneration of…
In this paper, we shall study the boundary case for complex Monge-Amp\`ere type equations under certain geometric assumptions.
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
Let $(X,\omega)$ be a compact Hermitian manifold and let $\{\beta\}\in H^{1,1}(X,\mathbb R)$ be a real $(1,1)$-class with a smooth representative $\beta$, such that $\int_X\beta^n>0$. Assume that there is a bounded $\beta$-plurisubharmonic…
The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of…
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…
We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\mathcal{E}_\chi (\Omega)$ in the general case ($\chi(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these…