Fractional Hardy-Sobolev L1-embedding per capacity-duality

Abstract

This paper explores essence of the fractional ( 0 < s < 1 ) Hardy-Sobolev L 1 -embedding ‖ u ‖ L n n − s ≲ ‖ ∇ + s u ‖ L 1 + ‖ ∇ − s u ‖ L 1 for all u ∈ I s ( C c ∞ ∩ H 1 ) via both capacity and duality based on the Riesz potential operator I s and the fractional differential couple ∇ ± s . Naturally, we discover that f ∈ I s ( [ H ˚ − s , 1 ] ⁎ ) if and only if there exists g → = ( g 1 , . . . , g n ) ∈ ( L ∞ ) n such that f = R → ⋅ g → = ∑ j = 1 n R j g j in the John-Nirenberg space BMO where R → = ( R 1 , . . . , R n ) is the vector-valued Riesz transform - this equivalence characterizes R → ⋅ ( L ∞ ) n in the Fefferman-Stein decomposition: BMO = L ∞ + R → ⋅ ( L ∞ ) n and indicates that I s ( [ H ˚ − s , 1 ] ⁎ ) is a solution to the Bourgain-Brezis problem under n ≥ 2 : “What are the function spaces X , W 1 , n ⊂ X ⊂ BMO , such that every F ∈ X has a decomposition F = ∑ j = 1 n R j Y j where Y j ∈ L ∞ ?”.