In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hyt\"onen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{ \infty}$ constants inspired in the Fujii-Wilson and Hrus\v{ c}\v{ e}v classical constants. These estimates have many new applications including mixed bounds for multilinear Calder\'on--Zygmund operators and their commutators with $BMO$ functions, square functions and multilinear Fourier multipliers.
Categories: math.CA