Borderline Weighted Estimates for Commutators of Fractional Integrals

Let $I_{\alpha,\vec{b}}$ be the multilinear commutators of the fractional integrals $I_{\alpha}$ with the symbol $\vec{b}=(b_1, \cdots,b_k )$. We show that the constant of borderline weighted estimates for $I_{\alpha}$ is $\frac{1}{{\varepsilon}}$, and for $I_{\alpha,{\vec{b}}}$ is $\frac{1}{{\varepsilon}^{k+1}}$ with each $b_i$ belongs to the Orlicz space $Osc_{\exp L^{s_i}}$.