Degenerate Poincar\'e-Sobolev inequalities

EZEQUIEL RELA

Buenos Aires

Workshop; Cuarta Escuela sobre Análisis Funcional y Geometría; 2018

IAM - CONICET

In this talk I will present some recents results on weighted weighted Poincar\'e and Poincar\'e-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form$$\left (\frac{1}{w(Q)}\int_Q|f-f_Q|^{q}w\right )^\frac{1}{q}\le C_w\ell(Q)\left (\frac{1}{w(Q)}\int_Q |\nabla f|^p w\right )^\frac{1}{p},$$with different quantitative estimates for both the exponent $q$ and the constant $C_w$.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$$\avgint_Q |f-f_Q| d\mu \le a(Q),$$for all cubes $Q\subset\mathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=\frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ 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. We also apply our method to obtain similar estimates in the scale of Lorentz spaces. This is a joint work with Carlos P\'erez Moreno from BCAM.