In this paper we first introduce a transform for convex functions and use it to prove a Bernstein theorem for a Monge-Amp`ere equation in half space.We then prove the optimal global regularity for a class of Monge-Amp`ere type equations arising in a number of geometric problems such as Poincar´e metrics, hyperbolic affine spheres, and Minkowski type problems.

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Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

The purpose of this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable