Necessary and sufficient conditions for the bounds of the Calderón type commutator for the Littlewood–Paley operator. (January 2016) - Chen, Yanping; Ding, Yong
Abstract: For b ∈ L loc 1 ( R n ), we define the Calderón type commutator for the Littlewood–Paley operator g function by g Ω, 1 ; b ( f ) ( x ) = ( ∫ 0 ∞ | 1 t 2 ∫ | x − y | ≤ t Ω ( x − y ) | x − y | n − 1 ( b ( x ) − b ( y ) ) f ( y ) d y | 2 d t t ) 1 / 2, which is a new kind of commutator of Littlewood–Paley operator g function. In this paper the authors obtain the necessary and sufficient conditions on the function b to guarantee that g Ω, 1 ; b is a bounded operator on L 2 ( R n ) . Namely, if Ω ∈ L ( log + L ) 1 / 2 ( S n − 1 ) and b ∈ L i p ( R n ), then g Ω, 1 ; b is bounded on L 2 ( R n ) . And the converse is: if g Ω, 1 ; b is bounded on L 2 ( R n ), then b ∈ L i p ( R n ) .