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.
In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \[\|\varphi\|_{L^p(v,\Omega)} \leq S(p,1) \| \sqrt{Q}\nabla \varphi\|_{L^p(\Omega)},\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\nabla u)\in QH^{1,p}_0(v,\Omega...
Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularit...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient...
We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}^q(\Omega)$, consisting of $L^q$-functions with Fourier support in a convex set $\Omega \subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely, \[ \int_{\Omega}\dfrac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}dx\leq C\|f\...
