Self-improving properties for abstract Poincar\'e type inequalities

TL;DR

This paper develops a unified abstract framework for self-improving properties of generalized Poincaré inequalities in Euclidean spaces, encompassing classical cases and those involving semigroups related to elliptic operators.

Contribution

It introduces a novel abstract setting that captures oscillations via operators with off-diagonal decay, unifying classical and semigroup-based Poincaré inequalities.

Findings

Unified theory for self-improving Poincaré inequalities

Application to semigroups associated with elliptic operators

Proof of John-Nirenberg inequality in new contexts

Abstract

We study self-improving properties in the scale of Lebesgue spaces of generalized Poincar\'e 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 Poincar\'e 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].