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Sharp Weighted Bounds for Multilinear fractional Maximal type Operators with Rough Kernels

In this paper, we will give the weighted bounds for multilinear fractional maximal type operators $\mathcal{M}_{Ω,α}$ with rough homogeneous kernels. We obtain a mixed $A_{(\vec{P},q)}-A_\infty$ bound and a $A_{\vec{P}}$ type estimate for $\mathcal{M}_{Ω,α}$. As an application, we give an almost sharp estimate for the multilinear fractional integral operator with rough kernels $\mathcal{I}_{Ω,α}$.