$A_p$-$A_{\mathrm{\infty }}$ estimates for multilinear maximal and sparse operators

We obtain mixed $A_p$–$A_{\mathrm{\infty }}$ estimates for a large family of multilinear maximal and sparse operators. Operators from this family are known to control for instance multilinear Calder'on–Zygmund operators, square functions, fractional integrals, and the bilinear Hilbert transform. Our results feature a new multilinear version of the Fujii–Wilson $A_{\mathrm{\infty }}$ characteristic that allows us to recover the best known estimates in terms of the $A_p$ characteristic for dependent weights as a special case of the mixed characteristic estimates for general tuples of weights.