New bounds for bilinear Calderón–Zygmund operators and applications

Abstract

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 $A_{\infty }$ constants inspired in the Fujii–Wilson and Hruscev classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón–Zygmund operators and their commutators with BMO functions, square functions and multilinear Fourier multipliers.