On Bloom-type characterizations of the higher-order commutators of Marcinkiewicz integrals
UDC 517.9 Let Ω be homogeneous of degree zero, have mean value zero, and integrable on the unit sphere. For mN, let bL_loc^1(R^n) and let the higher-order commutator of the Marcinkiewicz integral μ_Ω,b^m be defined by gather*μ_Ω,b^m(f)(x)=( _0^∞ | _|x-y| ≤ t Ω(x-y)|x-y|^n-1 [b(x)-b(y)]^mf(y) dy|^2 d tt^3 )^12 .gather* We establish a sparse domination of μ_Ω,b^m for ΩLip(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 gather*μ_Ω,α,b^*,m(f)(x)=( _R_+^n+1 (tt+|x-y|)^nα| _|y-z| ≤ t Ω(y-z)|y-z|^n-1 [b(x)-b(z)]^mf(z) dz|^2dydtt^n+3 )^12 ,α>1,gather* and gather*μ_Ω,S,b^m(f)(x)=( _|x-y|<t| _|y-z| ≤ t Ω(y-z)|y-z|^n-1 [b(x)-b(z)]^mf(z) dz|^2dydtt^n+3 )^12 .gather* , number=6, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Chen, Yanping and Tian, Tian, year=2024, month=Jul., pages=931–948