On the H\"older continuous subsolution problem for the complex Monge-Amp\`ere equation

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.