  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].
