In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón-Zygmund operators with Dini-continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple A∞ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón-Zygmund operators and their commutators with BM O functions, square functions and multilinear Fourier multipliers.

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Abstract. Let P = (p1, . . . , pm) with 1 < p1, . . . , pm < ∞, 1/p1 + · · · + 1/pm = 1/p and w = (w1, . . . , wm) ∈ A P . In this paper, we investigate the weighted bounds with dependence on aperture α for multilinear square functions S α,ψ ( f ). We show thatThis result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calderón-Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.