In this paper a new Alexandro - Bakelman - Pucci estimate is obtained for solutions of fully nonlinear uniform elliptic equations in any proper n-dimensional domain. The novelty, with respect to the existing literature, is that the estimate does not depend on the geometry of the domain and extends to solutions vanishing at infinity in unbounded domains. No decay condition on the first order coefficient is assumed, but instead a ''positive'' drift. The existence of solutions vanishing at infinity is also shown, based on the ABP estimates previously proved.
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