Lp self-improvement of generalized Poincaré inequalities in spaces of homogeneous type

In this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincaré inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincaré or Sobolev–Poincaré inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup e−tΔ with Δ being the Laplace–Beltrami operator. We obtain generalized Poincaré inequalities with oscillations that involve the semigroup e−tΔ and with right hand sides containing either ∇ or Δ1/2.