WEIGHTED ENDPOINT ESTIMATES FOR COMMUTATORS OF CALDERÓN-ZYGMUND OPERATORS

Let δ ∈ (0, 1] and T be a δ-Calderón-Zygmund operator. Let w be in the Muckenhoupt class A1+δ/n(Rn) satisfying ∫ R n w ( x ) 1 + | x | n d x < ∞ . When b ∈ BMO(Rn), it is well known that the commutator [b, T] is not bounded from H1(Rn) to L1(Rn) if b is not a constant function. In this article, the authors find out a proper subspace BMOw (Rn) of BMO(Rn) such that, if b ∈ BMOw (Rn), then [b, T] is bounded from the weighted Hardy space H w 1 ( R n ) to the weighted Lebesgue space L w 1 ( R n ) . Conversely, if b ∈ BMO(Rn) and the commutators of the classical Riesz transforms { [ b , R j ] } j = 1 n are bounded from H w 1 ( R n ) to L w 1 ( R n ) , then b ∈ BMOw (Rn).