Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space. (29th May 2014) - Zhou, Jiang; Li, Peng
Abstract : LetT σ be the multilinear Fourier multiplier operator associated with multiplierσ satisfying the Sobolev regularity thats u p l ∈ Z σ l W s 1, …, s m ( R m n ) < ∞ for somes k ∈ ( n / 2, n ]   ( k = 1, …, m ) . The authors prove that ifb 1, …, b m ∈ B M O ( R n ) andw ⃗ ∈ ∏ k = 1 m ‍ A p k / t k ( t k = n / s k ), then the commutatorT σ, Σ b is bounded fromL p 1 ( R n, w 1 ) × ⋯ × L p m ( R n, w m ) toL p ( R n, ν w ⃗ ) . Moreover, the authors also prove that ifb 1, …, b m ∈ V M O ( R n ) andw ⃗ ∈ ∏ k = 1 m ‍ A p k / t k ( t k = n / s k ), then the commutatorT σ, Σ b is compact operator fromL p 1 ( R n, w 1 ) × ⋯ × L p m ( R n, w m ) toL p ( R n, ν w ⃗ ) .