Degenerate Poincar\'e-Sobolev inequalities

TL;DR

This paper investigates weighted Poincaré and Poincaré-Sobolev inequalities, analyzing their dependence on weight constants, introducing a Sobolev exponent related to weights, and establishing optimal estimates and limitations for these inequalities.

Contribution

It provides explicit quantitative estimates for weighted Poincaré-Sobolev inequalities, introduces a Sobolev-type exponent for weights, and explores the limitations of such inequalities for certain weight classes.

Findings

Derived inequalities with explicit dependence on $A_p$ constants.

Introduced a Sobolev-type exponent $p^*_w$ related to weights.

Showed the optimality of estimates and limitations for $A_1$ weights.

Abstract

We study 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 $$ \frac{1}{|Q|}\int_Q |f-f_Q| d\mu \le a(Q), $$ 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 $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $p\geq1$, Poincar\'e inequality valid for the whole class of $RH_\infty$ weights by showing their intimate connection with the failure of Poincar\'e inequalities, $(p,p)$ in the range $0<p<1$.