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Let $X$ be a compact Kahler manifold of dimension $n$ and $omega$ a Kahler form on $X$. We consider the complex Monge-Amp`ere equation $(dd^c u+omega)^n=mu$, where $mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $omega$-plurisubharmonic function. We show that the equation admits a Holder continuous solution {it if and only if} the measure $mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is Holder continuous. A similar result is also obtained for the complex Monge-Amp`ere equations on domains of $mathbb{C}^n$.
We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp`ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in par
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations o
We develop a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel cite{GL21a} we h
Let $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure on $Omega$. We study the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. First we give a suffi
We study complex geodesics and complex Monge-Amp`{e}re equations on bounded strongly linearly convex domains in $mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domai