Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications

Abstract

Let 0<s<1 and p>1 be such that ps<N. Assume that Ω is a bounded domain containing the origin. Starting from the ground state inequality by R. Frank and R. Seiringer in [14] to obtain:(1)The following improved Hardy inequality for p⩾2:For all q<p, there exists a positive constant C≡C(Ω,q,N,s) such that∫RN∫RN|u(x)−u(y)|p|x−y|N+psdxdy−ΛN,p,s∫RN|u(x)|p|x|pdx⩾C∫Ω∫Ω|u(x)−u(y)|p|x−y|N+qsdxdy, for all u∈C0∞(RN). Here ΛN,p,s is the optimal constant in the Hardy inequality (1.2).(2)Define ps⁎=pNN−ps and let β<N−ps2, then∫RN∫RN|u(x)−u(y)|p|x−y|N+ps|x|β|y|βdydx⩾S(N,p,s,β)(∫RN|u(x)|ps⁎|x|2βps⁎pdx)pps⁎, for all u∈C0∞(Ω) where S≡S(N,p,s,β)>0.(3)If β≡N−ps2, as a consequence of the improved Hardy inequality, we obtain that for all q<p, there exists a positive constant C(Ω) such that∫RN∫RN|u(x)−u(y)|p|x−y|N+ps|x|β|y|βdydx⩾C(Ω)(∫Ω|u(x)|ps,q⁎|x|2βps,q⁎pdx)pps,q⁎, for all u∈C0∞(Ω) where ps,q⁎=pNN−qs. Notice that the previous inequalities can be understood as the fractional extension of the Callarelli–Kohn–Nirenberg inequalities in [8].