Poincaré, Sobolev and Rubio de Francia

EZEQUIEL RELA

Villa General Belgrano

Encuentro; XIV Encuentro Nacional de Analistas Alberto P. Calderón; 2018

CONICET - UBA

In this talk I will present some recents results on 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. 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 will also discuss an interesting application of a variation of the famous extrapolation technique of Rubio de Francia to a problem related to the Keith-Zhong theorem on the open ended condition for Poincar\'e inequalities.\ This is a joint work with Carlos P\'erez Moreno from BCAM, Bilbao, Spain.