Weighted Hardy Space Estimates for Commutators of Calderon-Zygmund Operators

Let delta is an element of (0,1] and T be a delta-Calderon-Zygmund operator. Let p is an element of (0,1] be such that p(1 + delta/n) > 1, and assume that w belongs to the Muckenhoupt weight class A(p(1+delta/n)) (R-n) with the property integral(Rn) w(x)/(1+vertical bar x vertical bar)(np) dx < infinity. When b is an element of BMO(R-n), it is well-know that the commutator [b, T] is not bounded from H-p(R-n) into L-p(R-n) if b is not a constant function. In this paper, we find a proper subspace BMOw,p(R-n) of BMO(R-n) such that, if b is an element of BMOw,p(R-n), then [b, T] is bounded from the weighted Hardy space H-w(p) (R-n) into the weighted Lebesgue space L-w(p)(R-n). Conversely, if b is an element of BMO(R-n) and the commutators {[b, R-j]}(j=1)(n) of the classical Riesz transforms are bounded from H-w(p) (R-n) into L-w(p)(R-n), then b is an element of BMOw,p(R-n).