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Transactions of the American Mathematical Society

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Self-improving properties for abstract Poincaré type inequalities

Authors: Frédéric Bernicot and José María Martell
Journal: Trans. Amer. Math. Soc. 367 (2015), 4793-4835
MSC (2010): Primary 46E35; Secondary 47D06, 46E30, 42B25, 58J35
Published electronically: November 12, 2014
MathSciNet review: 3335401
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Abstract: We study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the identity, semigroups or mean value operators) that have off-diagonal decay in some range. Our results provide a unified theory that is applicable to the classical Poincaré inequalities, and furthermore it includes oscillations defined in terms of semigroups associated with second order elliptic operators as those in the Kato conjecture. In this latter situation we obtain a direct proof of the John-Nirenberg inequality for the associated $ BMO$ and Lipschitz spaces of S. Hofmann, S. Mayboroda, and A. McIntosh.

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Additional Information

Frédéric Bernicot
Affiliation: Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2, Rue de la Houssinière F-44322 Nantes Cedex 03, France

José María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain

Keywords: Self-improving properties, BMO and Lipschitz spaces, John-Nirenberg inequalities, generalized Poincar\'e-Sobolev inequalities, pseudo-Poincar\'e inequalities, semigroups, dyadic cubes, weights, good-$\lambda$ inequalities.
Received by editor(s): March 21, 2013
Published electronically: November 12, 2014
Additional Notes: The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087
Both authors wish to thank Pascal Auscher, Steve Hofmann and Svitlana Mayboroda for helpful comments concerning some of the applications.
Article copyright: © Copyright 2014 American Mathematical Society

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