Endpoint Estimates For The Commutators Of Singular Integral Operators On Spaces Of Homogeneous Type

About the commutators of singular integral operators on the spaces of L^p(Rⁿ), R.Coifman etc proved that the commutator T_bf = [b,T]f = T(bf) -bT(f) is bounded on Lp(Rn)(l < p < ∞) if and only if b BMO, where T is a classical singular integral operator. C. Perez studied the endpoint estimates of commutators of singular integral operators and show that Tb, is neither weak-type(1,1) nor bounded from H1(Rn) to L1(Rn), but it satisfies an inequality of L(logL) type and is bounded from H1b(Rn] to L1 (Rn}.About the commutators of singular integral operators of spaces of homoge-neous type, Bramanti and Cerutti prove the boundedness of the commutators of singular integral operators with BMO functions on Lp(X)(1 < p < ∞).In this paper, we show that the commutators of singular integral operators with BMO functions on spaces of homogeneous type satisfy the inequality of L(logL) type, that isand extend this result to higher order commutators.