Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain

Abstract

The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space Rn are well-established, whereas only partial results are known for the non-smooth domains. In this paper, Ω is a non-smooth domain of Rn that is bounded and uniform. Suppose p, q∈[1,∞) and s∈(n(1p−1q)+,1) with n(1p−1q)+:=max{n(1p−1q),0}. The authors show that three typical types of fractional Triebel–Lizorkin spaces, on Ω: Fp,qs(Ω), F˚p,qs(Ω) and F˜p,qs(Ω), defined via the restriction, completion and supporting conditions, respectively, are identical if Ω is E-thick and supports some Hardy inequalities. Moreover, the authors show the condition that Ω is E-thick can be removed when considering only the density property Fp,qs(Ω)=F˚p,qs(Ω), and the condition that Ω supports Hardy inequalities can be characterized by some Triebel–Lizorkin capacities in the special case of 1≤p≤q<∞.