Maximal bilinear singular integral operators associated with dilations of planar sets

We obtain square function estimates and bounds for maximal singular integral operators associated with bilinear multipliers given by characteristic functions of dyadic dilations of certain planar sets. As a consequence, we deduce pointwise almost everywhere convergence for lacunary partial sums of bilinear Fourier series with respect to methods of summation determined by the corresponding planar sets. © 2006 Elsevier Inc. All rights reserved