Where Sobolev interacts with Gagliardo–Nirenberg

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