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We study weak quasi-plurisubharmonic solutions to the Dirichlet problem forthe complex Monge-Am\`ere equation on a general Hermitian manifold withnon-empty boundary. We prove optimal subsolution theorems: for bounded andH\"older continuous quasi-plurisubharmonic functions. The continuity of thesolution is proved for measures that well dominated by capacity, for examplemeasures with $L^p$, $p>1$ densities, or moderate measures in the sense ofDinh-Nguyen-Sibony.

The Dirichlet problem for the Monge-Amp\`ere equation on Hermitian manifolds with boundary