Abstract
We investigate the validity of the fractional Gagliardo-Nirenberg-Sobolev inequality(1)‖f‖Wr,q(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ,∀f∈Ws1,p1(Ω)∩Ws2,p2(Ω).
Here, s1,s2,r are non-negative numbers (not necessarily integers), 1≤p1,p2,q≤∞, and we assume, for some θ∈(0,1), the standard relations(2)r<s:=θs1+(1−θ)s2 and 1q=(θp1+1−θp2)−s−rN.
Formally, estimate (1) is obtained by combining the “pure” fractional Gagliardo-Nirenberg style interpolation inequality(3)‖f‖Ws,p(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ(with 1/p:=θ/p1+(1−θ)/p2) with the fractional Sobolev style embedding(4)Ws,p(Ω)↪Wr,q(Ω),0≤r<s,1≤p<q≤∞,1q=1p−s−rN,p(s−r)≤N.
Estimates (3) and (4) are true “most of the time”, but not always; the exact range of validity of (3) and (4) has been known. Combining these results, we infer that (1) is valid “most of the time”. However, the validity of (1) when (3) and/or (4) fail was unclear. The goal of this paper is to characterize the values of s1,s2,r,p1,p2,q,θ,N such that (1) holds.