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

Jun Cao, Yongyang Jin, Yuanyuan Li and Qishun Zhang
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Jun Cao: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
Yongyang Jin: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
Yuanyuan Li: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
Qishun Zhang: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China

Mathematics, 2022, vol. 10, issue 4, 1-25

Abstract: 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 < ∞ .

Keywords: Triebel–Lizorkin space; Hardy inequality; uniform domain; fractional Laplacian (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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