On uniform estimate in Calabi-Yau theorem

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We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampère equation.

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On the constant scalar curvature Kähler metrics (I)—A priori estimates (2017, methodology): “This argument is due to Blocki (c.f. [2]).”
Complex Monge-Ampere equations and totally real submanifolds (2009, methodology): “equently simpliﬁed by Kazdan [44] for n= 2, and by Aubin [1] and Bourguignon independently for arbitrary dimension (see e.g. [58] and [64]). Alternative proofs were given by Kolodziej [45] and Blocki [8] based on the pluripotential theory ([6]) and the L2 stability of the complex Monge-Amp`ere opera”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “Following Blocki [1] and Székelyhidi [50], we obtain C0 estimates from the Alexandrov-BakelmanPucci (ABP) inequality, using this time a parabolic version of ABP due to K. Tso [62].”
A gradient estimate in the Calabi–Yau theorem (2009, background): “In fact, he improved this estimate as follows: for any p > 1 the constant C0 depends on M, ||f||Lp(M) and p (see also [ B4 ]).”