Let be a real polynomial of degree defined by
The Radon transforms , and of are defined by
respectively, where is a cutoff function. Let be the closed convex hull of the points , , , , , and . When is a homogeneous polynomial, and and , Phong and Stein proved that is bounded from to , if is in the set minus the half-open segments and . They also proved that for to be bounded from to , it is necessary that is in , and in the case it is known that is bounded precisely for . The author gives a positive result for endpoint estimates in the case , and more. His main result is (for not necessarily homogeneous polynomials): if , then there is such that
where is independent of the coefficients . Using this, he gets endpoint estimates, not treated by Phong and Stein, in the homogeneous polynomial case.