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We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Amère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony., 38 pages, v2 final version incorporated the referee report

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony., Comment: 38 pages, v2 final version incorporated the referee report

Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a K\"ahler 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 H\"older continuous solution {\it if and only if} the measure $\mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is H\"older continuous. A similar result is also obtained for the complex Monge-Amp\`ere equations on domains of $\mathbb{C}^n$., Comment: 16 pages. Final version, to appear in Bulletin of the London Mathematical Society

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 particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting H\"older continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen., Comment: 18 pages

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp��re 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 particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting H��lder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen., 18 pages

We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp\`ere equation has the continuous solution. The modulus of continuity of the solution is also given if the right hand side is locally dominated by capacity., Comment: 8 pages

We solve the Dirichlet problem for the complex Monge-Amp��re equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Amp��re measure of a H��lder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If the boundary is H��lder continuous, then the solution is also H��lder continuous. In particular, the answer to a question of A. Zeriahi is always affirmative., 17 pages

We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp��re equation has the continuous solution. The modulus of continuity of the solution is also given if the right hand side is locally dominated by capacity., 8 pages

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions., Comment: 11 pages. Final version to appear in Ann. Inst. Fourier

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Amp\`ere equation admits H\"older continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Amp\`ere total mass, we obtain an affirmative answer to a question of Zeriahi., Comment: 16 pages

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Amp��re equation admits H��lder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Amp��re total mass, we obtain an affirmative answer to a question of Zeriahi., 16 pages

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions., 11 pages. Final version to appear in Ann. Inst. Fourier

We prove the existence and uniqueness of continuous solutions to the complex Monge-Amp\`ere type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a priori} are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove \cite{TW12a} and on a weak form of a conjecture of Demailly and Paun \cite{DP04}., Comment: 35 pages

We prove the existence and uniqueness of continuous solutions to the complex Monge-Amp��re type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a priori} are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove \cite{TW12a} and on a weak form of a conjecture of Demailly and Paun \cite{DP04}., 35 pages

We prove the subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in $\bC^n$., 18 pages. Corrected typos. To appear in Universitatis Iagellonicae Acta Mathematica