Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Real analysis related to the Monge-Ampère equation
HTML articles powered by AMS MathViewer

by Luis A. Caffarelli and Cristian E. Gutiérrez PDF
Trans. Amer. Math. Soc. 348 (1996), 1075-1092 Request permission


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 $\mathcal {F}.$
Similar Articles
Additional Information
  • Luis A. Caffarelli
  • Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
  • MR Author ID: 44175
  • Email:
  • Cristian E. Gutiérrez
  • Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
  • Email:
  • Received by editor(s): December 23, 1994
  • Received by editor(s) in revised form: January 24, 1995
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1075-1092
  • MSC (1991): Primary 35J60, 42B20; Secondary 35B45, 42B25
  • DOI:
  • MathSciNet review: 1321570