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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Real analysis related to the Monge-Ampère equation

Authors: Luis A. Caffarelli and Cristian E. Gutiérrez
Journal: Trans. Amer. Math. Soc. 348 (1996), 1075-1092
MSC (1991): Primary 35J60, 42B20; Secondary 35B45, 42B25
MathSciNet review: 1321570
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Abstract: In this paper we consider a family of convex sets in $\mathbf {R}^{n}$, $\mathcal {F}= \{S(x,t)\}$, $x\in \mathbf {R}^{n}$, $t>0$, satisfying certain axioms of affine invariance, and a Borel measure $\mu$ satisfying a doubling condition with respect to the family $\mathcal {F}.$ 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 $\mathcal {F}.$ This is achieved by showing first a Besicovitch-type covering lemma for the family $\mathcal {F}$ and then using the doubling property of the measure $\mu .$ 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 $\mathcal {F}.$

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Additional Information

Luis A. Caffarelli
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
MR Author ID: 44175

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Keywords: Convex sets, real Monge-Ampère equation, covering lemmas, real-variable theory, BMO
Received by editor(s): December 23, 1994
Received by editor(s) in revised form: January 24, 1995
Article copyright: © Copyright 1996 American Mathematical Society