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We give two applications of the Aleksandrov-Bakelman-Pucci estimate to theCalabi-Yau equation on symplectic four-manifolds. The first is solvability ofthe equation on the Kodaira-Thurston manifold for certain almost-Kahlerstructures assuming $S^1$-invariance, extending a result ofBuzano-Fino-Vezzoni. The second is to reduce the general case of Donaldson'sconjecture to a bound on the measure of a superlevel set of a scalar function.

The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation