  The so-called triangular Hilbert transform is an elegant trilinear singular
integral form which specializes to many well studied objects of harmonic
analysis. We investigate $L^p$ bounds for a dyadic model of this form in the
particular case when one of the functions on which it acts is essentially
one-dimensional. This special case still implies dyadic analogues of
boundedness of the Carleson maximal operator and of the uniform estimates for
the one-dimensional bilinear Hilbert transform.
