Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

We prove that the commutator $[b,I_\alpha]$, $b\in \mathit{BMO}$, $I_\alpha$ the fractional integral operator, satisfies the sharp, modular weak-type inequality
$$
|\{ x \in \mathbb{R}^n : |[b,I_\alpha]f(x)| > t \}| \leq C\,\Psi \left(\int_{{\mathbb R}^n} B \left(\|b\|_{\mathit{BMO}} \frac{|f(x)|}{t}\right)\,dx\right),
$$
where $B(t)=t\log(e+t)$ and $\Psi(t)=[t\log(e+t^{\alpha/n})]^{n/(n-\alpha)}$. These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality,
$$
M^\#([b,I_\alpha]f)(x) \leq C\|b\|_{\mathit{BMO}}[ I_\alpha f(x) + M_{\alpha,B}f(x)],
$$
where $M^\#$ is the sharp maximal operator, and $M_{\alpha,B}$ 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 $M_{\alpha,B}$ which are of interest in their own right.