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Joint works with S. Dinew and Nguyen Ngoc Cuong.
Let \((X,\omega)\) be a compact Hermitian manifold of complex dimension \(n\). We study the
weak solutions to the complex Monge-Amp\`ere equation
\[
(\omega + dd^c \varphi)^n = f \omega^n, \quad
\omega + dd^c \varphi \geq 0,
\]
where \(0\leq f \in L^p(X, \omega^n)\), \(p>1\), and \(dd^c=\frac{i}{\pi}\partial \bar\partial\), with the inequality understood in the
sense of currents.
The main results include a priori estimates and the existence of continuous solutions of the complex Monge-Amp\`ere equation
with the right hand side in \(L^p , p>1\).
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