A Bilinear Sparse Domination for the Maximal Singular Integral Operators with Rough Kernels

Let Ω be homogeneous of degree zero, integrable on S d - 1 and have mean value zero, T Ω be the homogeneous singular integral operator with kernel Ω ( x ) | x | d and T Ω ∗ be the maximal operator associated to T Ω . In this paper, the authors prove that if Ω ∈ L ∞ ( S d - 1 ) , then for all r ∈ ( 1 , ∞ ) , T Ω ∗ enjoys a ( L Φ , L r ) bilinear sparse domination with Φ ( t ) = t log log ( e 2 + t ) . Some applications of this bilinear sparse domination are also given.