Degenerate Poincar\'e inequalities

EZEQUIEL RELA

Porto Alegre

Congreso; ICM 2018 Satellite Conference Harmonic Analysis; 2018

UFRGS

In this talk I will present some recents results on weighted weightedPoincare and Poincare-Sobolev type inequalities with an explicit analysison the dependence on the Ap constants of the involved weights. We obtaininequalities of the form1w(Q)ZQjf 􀀀 fQjqw1q Cw`(Q)1w(Q)ZQjrfjpw1p;with dierent quantitative estimates for both the exponent q and the con-stant Cw. We will derive those estimates together with a large variety ofrelated results as a consequence of a general selmproving property sharedby functions satisfying the inequality􀀀ZQjf 􀀀 fQjd a(Q);for all cubes Q Rn and where a is some functional that obeys a specicdiscrete geometrical summability condition. We introduce a Sobolev-typeexponent pw > p associated to the weight w and obtain further improve-ments involving Lpw norms on the left hand side of the inequality above.For the endpoint case of A1 weights we reach the classical critical Sobolevexponent p = pnn􀀀p which is the largest possible and provide dierent type ofquantitative estimates for Cw. We also show that this best possible estimatecannot 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 fur-ther extensions to two weights Poincare inequalities and to the case of higherorder derivatives. We also apply our method to obtain similar estimates inthe scale of Lorentz spaces.This is a joint work with Carlos Perez Moreno from BCAM.