Fractional Hardy-Sobolev inequalities for canceling elliptic differential operators

Abstract

Let A(D) be an elliptic homogeneous linear differential operator of order ν on R N , N ≥ 2, from a complex vector space E to a complex vector space F. In this paper we show that if ℓ ∈ R satisfies 0 < ℓ < N and ℓ ≤ ν, then the estimateis canceling in the sense of V. Schaftingen [13]. Here (−∆) 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 [2]. We also present a local version of the previous inequality for operators with smooth variables coefficients.