Endpoint Estimates For Commutators Of Singular Integral Operator With Dini-type Kernel

On the commutators of singular integral operators with standard kernels with BMO functions, R.Coifman, R.Rochberg and G.Weiss [2] proved that b G BMO is sufficient for the commutator operator [b, T]f = T(bf) - bT(f) to be bounded on L^p(Rⁿ), 1 < p < ∞. C.Perez studied the endpoint estimates of commutators.Let T be a singular integral operator with Dini-type kernal as follow:where Ω satisfies:(i). Ω(λx) = Ω(x), λ> 0; (ii). ∫(Sn₁) Ω(x)dσ(x) = 0; (iii). Dini-type condition:whereω(σ) = sup{|Ω(x) - Ω(y)|:|x - y| < σ, x,y∈ S^n-1}.The commutators [b, T] to be considered in this paper are composed of singular integral operators with Dini-type kernel defined as above and BMO functions. The purpose is to provide endpoint estimates for these commutator operators.In this paper, we show that the commutators [b, T] satisfies L(logL) inequalities,and is bounded from H1(Rⁿ) to weak L¹(Rⁿ), and is bounded from H_b¹(Rn) toL¹(Rⁿ), and extend these results to higher order commutators.