A remark on the fractional Hardy inequality with a remainder term

Abstract

We prove in this note the following sharpened fractional Hardy inequality:LetN⩾1, 0<s<1, N>2s, andΩ⊂RNa bounded domain. Then for all1<q<2, there exists a positive constantC=C(Ω,q,N,s)such that for allu∈C0∞(Ω),(1)aN,s∫RN∫RN(u(x)−u(y))2|x−y|N+2sdxdy−ΛN,s∫RNu2(x)|x|2sdx⩾C(Ω,q,N,s)∫Ω∫Ω(u(x)−u(y))2|x−y|N+qsdxdy,whereaN,s=22s−1π−N2Γ(N+2s2)|Γ(−s)|andΛN,s=22sΓ2(N+2s4)Γ2(N−2s4).