A remark on the fractional Hardy inequality with a remainder term

We prove in this note the following sharpened fractional Hardy inequality: Let N >= 1, 0 < s < 1, N > 2s, and Omega subset of R-N a bounded domain. Then for all 1 < q < 2, there a positive constant C = C (Omega, q, N, s) such that for all u is an element of C-0(infinity) (Omega) a(N,S) integral(RN) integral(RN) (u(x) - u(y))(2)/vertical bar x - y vertical bar(N+2s) dxdy -Lambda(N,S) integral(RN) u(2)(x)/vertical bar x vertical bar(2s) dx >= C(Omega,q,N,s) integral(Omega) integral(Omega) (u(x) - u(y))(2)/vertical bar x - y vertical bar(N+qs) dxdy, (1) where a(N,s) = 2(2s-1) pi(-N/2) Gamma(N+2s/2)/vertical bar Gamma(-s) vertical bar and Lambda(N,s) =2(2s) Gamma(2)(N+2s/4)/Gamma(2) (N-2s/4) (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.