Fractional Hardy-Sobolev inequalities for canceling elliptic differential operators

Abstract

Let $A(D)$ be an elliptic homogeneous linear differential operator of order $\nu$ on $\mathbb{R}^{N}$, $N \geq 2$, from a complex vector space E to a complex vector space F. In this paper we show that if $\ell\in \mathbb{R}$ satisfies $0< \ell <N$ and $\ell \leq \nu$, then the estimate \begin{equation}\nonumber \left(\int_{\mathbb{R}^{N}}| (-\Delta)^{(\nu-\ell)/2}u(x)|^{q}|x|^{-N+(N-\ell)q}\,dx\right)^{1/q}\leq C \|A(D)u\|_{L^{1}} \end{equation} holds for every $u \in C_{c}^{\infty}(\mathbb{R}^{N};E)$ and $1\le q<\frac{N}{N-\ell}$ if and only if $A(D)$ is canceling in the sense of V. Schaftingen [VS]. Here $(-\Delta)^{a/2}u$ is the fractional Laplacian defined as a Fourier multiplier. This estimate extends, implies and unifies a series of classical inequalities discussed by P. Bousquet and V. Schaftingen in [BVS]. We also present a local version of the previous inequality for operators with smooth variables coefficients.}