Regularity of Solutions to Monge-Amp\`ere Equations on Stein Spaces

In this note, we study degenerate complex Monge-Ampère equations on singular Stein spaces with right-hand sides depending on the unknown.

First, we prove continuity up to the boundary for solutions.

Next, under a holomorphic peak-set condition, we prove local Hölder estimates on the regular locus, allowing the boundary datum to fail to be Hölder continuous at the singular boundary.

This includes the case of finite singular boundary sets.