Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups
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The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincaré type inequalities. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincaré inequalities.
KeywordsSemigroups Heat kernels Self-improving properties Generalized Poincaré–Sobolev and Hardy inequalities Pseudo-Poincaré inequalities Dyadic cubes Weights Good-λ inequalities
Mathematics Subject Classifications (2010)46E35 (47D06, 46E30, 42B25)
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