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