On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

Abstract

For N≥3 and p∈(1,N), we look for g∈Lloc1(RN) such that the following weighted logarithmic Sobolev inequality:∫RNg|u|plog⁡|u|pdx≤γlog⁡(C(g,γ)∫RN|∇u|pdx), holds true for all u∈D01,p(RN) with ∫RNg|u|pdx=1, for some γ,C(g,γ)>0. For each r∈(p,NpN−p], we identify a Banach function space Hp,r(RN) such that the above inequality holds for g∈Hp,r(RN). For γ>rr−p, we also find a class of g for which the best constant C(g,γ) in the above inequality is attained in D01,p(RN). Further, for a closed set E with Assouad dimension =d<N and a∈(−(N−d)(p−1)p,(N−p)(N−d)Np), we establish the following logarithmic Hardy inequality∫RN|u|pδEp(a+1)log⁡(δEN−p−pa|u|p)dx≤Nplog⁡(C∫RN|∇u|pδEpadx), for all u∈Cc∞(RN) with ∫RN|u|pδEp(a+1)dx=1, for some C>0, where δE(x) is the distance between x and E. The second order extension of the logarithmic Hardy inequality is also obtained.