Real analysis related to the Monge-Ampère equation |
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Authors: | Luis A. Caffarelli Cristian E. Gutié rrez |
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Affiliation: | School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 ; Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 |
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Abstract: | In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to |
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Keywords: | Convex sets real Monge-Amp`{e}re equation covering lemmas real-variable theory {em BMO} |
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