| In this dissertation we examine the regularity properties of Aleksandrov solutions to the Monge-Ampère equation det D2u = μ, where the Borel measure p satisfies a weak condition, D&epsis;, on the sections of u. The condition referred to is actually a family of conditions, indexed by &epsis; ∈ (0,1]. The case &epsis; = 1 corresponds to a doubling property. The doubling condition implies D&epsis; for every &epsis;. We show that when the function u is globally defined and its Monge-Ampère measure Mu is D &epsis;, then Mu is actually doubling, so that the conditions are equivalent in this case. We then explore the regularity properties of functions u defined on bounded domains for which Mu is D&epsis;. These results are an extension of the regularity theory available when 0 < λ ≤ μ ≤ Λ to a wider class of measures. |
