Let be a homogeneous polynomial of degree , and consider the oscillatory integral defined by
where (resp. ) denotes a smooth compactly supported function on (resp. ). The operator is closely related to a class of Radon transforms which we describe as follows: The phase gives rise to a family of curves in the plane via
with curves and suitable densities defined on them; the associated Radon transform is given by In this paper the authors give necessary and sufficient conditions for optimal estimates of the operator . Next, by viewing as a pseudo-differential operator whose symbol is the operator-valued function they deduce the boundedness and Sobolev regularity of the transform .