Degenerate Sobolev and Poincar\'e inequalities via extrapolation
In this paper we prove matrix weighted Sobolev and Poincar\'e inequalities using techniques derived from the theory of Rubio de Francia extrapolation. Given weights $w,\,v$ and a symmetric non-negative definite matrix valued function $Q$ defined on a connected open subset $\Omega$ of $\mathbb{f}R^n$ that satisfies the lower ellipticity condition \[ w(x)^p \leq |\sqrt{Q(x)}\xi|^p,\quad \xi\in \mathbb{R}^n, \] we give Lebesgue integrability conditions on the weights $w,v$ that ensure there exists $\tau\geq 1$ so that Sobolev and Poincar\'e inequalities of the form \[\bigg(\int_\Omega |u|^{\tau p} \,vdx\bigg)^{\frac{1}{\tau p}} \leq C(v,w) \bigg(\int_\Omega |\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and}\] \[\bigg(\int_\Omega |u-\langle u\rangle_{\Omega,v}|^{\tau p} \,v dx\bigg)^\frac{1}{\tau p} \leq C(v,w)\bigg(\int_\Omega|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}}\] hold for smooth $u$. We explore these and related results in the context of several examples that include John domains, the Heisenberg group, and CR manifolds.