  We consider bilinear multipliers that appeared as a distinguished particular
case in the classification of two-dimensional bilinear Hilbert transforms by
Demeter and Thiele [9]. In this note we investigate their boundedness on
Sobolev spaces. Furthermore, we study structurally similar operators with
symbols that also depend on the spatial variables. The new results build on the
existing L^p estimates for a paraproduct-like operator previously studied by
the authors in [5] and [10]. Our primary intention is to emphasize the
analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential
operators of order 0.
