On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals

Abstract

UDC 517.9 Let Ω be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere.  For m ∈ ℕ , let and let the higher-order commutator of the  Marcinkiewicz integral μ Ω , b m be defined by  μ Ω , b m ( f ) ( x ) = ( ∫ 0 ∞ | ∫ | x - y | ≤ t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] m f ( y ) ⅆ y | 2 ⅆ t t 3 ) 1 2 .  We establish a sparse domination of μ Ω , b m for  Ω ∈ L i p ( 𝕊 n - 1 ) . Moreover, we also give Bloom-type characterizations of the two-weighted boundedness of the higher-order commutators μ Ω , b m , μ Ω , α , b * , m , and μ Ω , S , b m , where  the higher-order commutators μ Ω , α , b * , m and μ Ω , S , b m are defined, respectively, by  μ Ω , α , b * , m ( f ) ( x ) = ( ∫ ∫ ℝ + n + 1 ( t t + | x - y | ) n α | ∫ | y - z | ≤ t Ω ( y - z ) | y - z |