Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

We prove that the commutator [b, I-alpha], b is an element of BMO, I-alpha the fractional integral operator, satisfies the sharp, modular weak-type inequality graphic where B(t) = t log(e + t) and Psi(t) = [tlog(e + t(alpha/n))](n/(n-a)). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M-# ([b.I-alpha]f)(x) less than or equal to C parallel tobparallel to(BMO)[I(alpha)f(x) + M-alpha,M-B f(x)], where M-# is the sharp maximal operator, and M-alpha,M-B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator to do so we prove one and two-weight norm inequalities for M-alpha,M-B which are of interest in their own right.