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We consider Monge-Amp\'ere equations with the right hand side function closeto a constant and from a function class that is larger than any H\"older classand smaller than the Dini-continuous class. We establish an upper bound for themodulus of continuity of the solution's second derivatives. This bound dependsexponentially on a quantity similar to but larger than the Dini semi-norm. Weestablish explicit control on the shape of the sequence of shrinking sections,hence revealing the nature of such exponential dependence.

An Interior A Priori Estimate for Solutions to Monge-Amp\`ere Equations with Right-Hand Side Close to One