Degenerate Poincare-Sobolev inequalities
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Abstract. We study weighted Poincar ́e and Poincar ́e-Sobolev type in- equalities with an explicit analysis on the dependence on the Ap con- stants of the involved weights. We obtain inequalities of the form with different quantitative estimates for both the exponent q and the constant Cw . We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality for all cubes Q ⊂ Rn and where a is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev- type exponent associated to the weight w and obtain further improvementsinvolvingLp∗w normsonthelefthandsideoftheinequality above. For the endpoint case of A1 weights we reach the classical critical Sobolev exponent p∗ = pn which is the largest possible and provide n−p different type of quantitative estimates for Cw. We also show that this best possible estimate cannot hold with an exponent on the A1 constant smaller than 1/p. As a consequence of our results (and the method of proof) we obtain further extensions to two weights Poincar ́e inequalities and to the case of higher order derivatives. Some other related results in the spirit of the work of Keith and Zhong on the open ended condition of Poincar ́e inequality are obtained using extrapolation methods. We also apply our method to obtain similar estimates in the scale of Lorentz spaces. We also provide an argument based on extrapolation ideas showing that there is no (p,p), p ≥ 1, Poincar ́e inequality valid for the whole class of RH∞ weights by showing their intimate connection with the failure of Poincar ́e inequalities, (p, p) in the range 0 < p < 1.