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
DOI:
https://doi.org/10.1090/S0002-9947-96-01473-0
MathSciNet review:
1321570
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
- [Ca1] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611, https://doi.org/10.2307/1971480
- [Ca2] Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965–969. MR 1127042, https://doi.org/10.1002/cpa.3160440809
- [Ca3] Luis A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math. 45 (1992), no. 9, 1141–1151. MR 1177479, https://doi.org/10.1002/cpa.3160450905
- [St] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
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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:
https://doi.org/10.1090/S0002-9947-96-01473-0
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