Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators (Q2258140)

In this paper, the authors obtain \(A_p\)-\(A_{\infty}\)-type sharp weighted bounds for multilinear maximal functions. Namely, they prove that for \(1<p_i<\infty\), \(i=1,\cdots,m\), and \(\frac{1}{p}=\frac{1}{p_1}+\cdots+\frac{1}{p_m}\) the estimate \[ \|\mathcal{M}(\vec{f})\|_{L^{p}(\vec{v}_{w})}\leq C_{n,m}[\vec{w}]^{\frac{1}{p}}_{A_{\vec{P}}}\prod\limits_{i=1}^{m}([\sigma_i]_{A_{\infty}})^{\frac{1}{p_i}}\prod\limits_{i=1}^{m}\|f_i\|_{L^{p_i}(w_i)} \] holds if \(\vec{w}\in A_{\vec{P}}\), where \(\sigma_{i}=w_{i}^{1-p^{\prime}}\), \(i=1,\cdots,m\). Moreover, the exponents are sharp. The authors also obtain the bound for mullilinear Calderón-Zygmund operators in terms of dyadic positive operators and the multilinear version of the \(A_2\) conjecture.