COMPACTNESS OF THE COMMUTATOR OF BILINEAR FOURIER MULTIPLIER OPERATOR

Letb 1,b 2∈ CMO(ℝ n ) andΤσ be the bilinear Fourier multiplier operator with associated multiplierσsatisfies the Sobolev regularity that sup κ ∈ ℤ ‖ σ κ ‖ W s 1 , s 2 ( ℝ 2 n ) < ∞ for somes 1,s 2∈ (n/2,n]. In this paper, it is proved that the commutator defined by T σ , b → ( f 1 , f 2 ) ( x ) = b 1 ( x ) T σ ( f 1 , f 2 ) ( x ) − T σ ( b 1 , f 1 , f 2 ) ( x ) + b 2 ( x ) T σ ( f 1 , f 2 ) ( x ) − T σ ( f 1 , b 2 f 2 ) ( x ) is a compact operator from L p 1 ( ℝ n ) × L p 2 ( ℝ n ) toLp (ℝ n ) whenpk ∈ (n/sk , ∞) (k= 1, 2),p∈ (1, ∞) with 1/p= 1/p 1+ l/p 2. 2010Mathematics Subject Classification: 42B15, 42B20. Key words and phrases: Bilinear Fourier multiplier, Commutator, CMO(ℝ n ), Compact operator.