The partial integrals of the N-fold Fourier integrals connected with elliptic polynomials (not necessarily
homogeneous; principal part of which has a strictly convex level surface) are considered. It is proved that if a + s > (N – 1)/2 and ap = N then the Riesz means of the nonnegative order s of the N-fold Fourier integrals of continuous finite functions from the Sobolev spaces Wpa (RN) converge uniformly on every compact set, and if a + s > (N – 1)/2 and ap = N, then for any x0 ∈ RN there exists a continuous finite function from the Sobolev space such, that the corresponding Riesz means of the N-fold Fourier integrals diverge to infinity at x0. AMS 2000 Mathematics Subject Classifications: Primary 42B08; Secondary 42C14.