Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space

Abstract EN

Let T σ be the multilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that s u p l ∈ Z σ l W s 1 , … , s m ( R m n ) < ∞ for some s k ∈ ( n / 2 , n ] ( k = 1 , … , m ) .

The authors prove that if b 1 , … , b m ∈ B M O ( R n ) and w ⃗ ∈ ∏ k = 1 m A p k / t k ( t k = n / s k ) , then the commutator T σ , Σ b is bounded from L p 1 ( R n , w 1 ) × ⋯ × L p m ( R n , w m ) to L p ( R n , ν w ⃗ ) .

Moreover, the authors also prove that if b 1 , … , b m ∈ V M O ( R n ) and w ⃗ ∈ ∏ k = 1 m A p k / t k ( t k = n / s k ) , then the commutator T σ , Σ b is compact operator from L p 1 ( R n , w 1 ) × ⋯ × L p m ( R n , w m ) to L p ( R n , ν w ⃗ ) .