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

The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space R n are well-established, whereas only partial results are known for the non-smooth domains. In this paper, Ω is a non-smooth domain of R n that is bounded and uniform. Suppose p , q ∈ [ 1 , ∞ ) and s ∈ ( n ( 1 p − 1 q ) + , 1 ) with n ( 1 p − 1 q ) + : = max { n ( 1 p − 1 q ) , 0 } . The authors show that three typical types of fractional Triebel–Lizorkin spaces, on Ω : F p , q s ( Ω ) , F ˚ p , q s ( Ω ) and F ˜ p , q s ( 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 F p , q s ( Ω ) = F ˚ p , q s ( 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 < ∞ .