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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 acomplex 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 $u \inC_{c}^{\infty}(\mathbb{R}^{N};E)$ 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 inequalitiesdiscussed by P. Bousquet and V. Schaftingen in [BVS]. We also present a localversion of the previous inequality for operators with smooth variablescoefficients.}

Fractional Hardy-Sobolev inequalities for canceling elliptic differential operators