  This is an introduction to a particular class of auxiliary complex
Monge-Amp\`ere equations which had been instrumental in $L^\infty$ estimates
for fully non-linear equations and various questions in complex geometry. The
essential comparison inequalities are reviewed and shown to apply in many
contexts. Adapted to symplectic geometry, with the auxiliary equation given now
by a real Monge-Amp\`ere equation, the method gives an improvement of an
earlier theorem of Tosatti-Weinkove-Yau, reducing Donaldson's conjecture on the
Calabi-Yau equation with a taming symplectic form from an exponential bound to
an $L^1$ bound.
