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Transactions of the American Mathematical Society
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
Email: caffarel@math.ias.edu

Cristian E. Gutiérrez
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: gutier@euclid.math.temple.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01473-0
PII: S 0002-9947(96)01473-0
Keywords: Convex sets, real Monge-Amp\`{e}re equation, covering lemmas, real-variable theory, {\em 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