Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\mathcal{F}.$

References [Enhancements On Off] (What's this?)

  • [Ca1] L. A. Caffarelli, Interior a Priori Estimates for Solutions of Fully Non-linear Equations, Annals of Math. 130 (1989), 189-213. MR 90i:35046
  • [Ca2] ------, Some Regularity Properties of Solutions of Monge-Ampère Equation, Comm. on Pure and App. Math. XLIV (1991), 965-969. MR 92h:35088
  • [Ca3] ------, Boundary Regularity of Maps with Convex Potentials, Comm. on Pure and App. Math. XLV (1992), 1141-1151. MR 93k:35054
  • [St] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Series #43, Princeton U. Press, Princeton, NJ, 1993. MR 95c:42002

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35J60, 42B20, 35B45, 42B25

Retrieve articles in all journals with MSC (1991): 35J60, 42B20, 35B45, 42B25

Additional Information

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

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

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