  In this paper we revisit our previous study of the local theory of prescribed
Jacobian equations associated with generating functions, which are extensions
of cost functions in the theory of optimal transportation. In particular, as
foreshadowed in the earlier work, we provide details pertaining to the
relaxation of a monotonicity condition A4w in the underlying convexity theory
and the consequent classical regularity. Taking advantage of recent work of
Kitagawa and Guillen, we also extend our classical regularity theory to the
weak form A3w of the critical matrix convexity conditions.
