Self-improving properties for abstract Poincaré type inequalities

Frederic Bernicot (1) , José Maria Martell (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 ICMAT - Instituto de Ciencias Matemàticas [Madrid]

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 [HMay,HMM].

Mots clés: Self-improving properties; BMO and Lipschitz spaces; John-Nirenberg inequalities; generalized Poincaré-Sobolev inequalities; pseudo-Poincaré inequalities; semigroups

Domaines: Analyse classique [math.CA]

Frederic Bernicot, José Maria Martell. Self-improving properties for abstract Poincaré type inequalities. Transactions of the American Mathematical Society, 2015, 367, pp.4793-4835. ⟨hal-00607963⟩