Abstract. The partial integrals of the $N$-fold Fourier integrals connected with elliptic polynomials (with 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 $W_p^a(R^N)$ converge uniformly on every compact set, and if $a+s=(N-1)/2$, $ap=N$, then for any $x_0\in R^N$ there exists a continuous finite function from the Sobolev space $W_p^a(R^N)$ such that the corresponding Riesz means of the $N$-fold Fourier integrals diverge to infinity at $x_0$.

AMS Subject Classification (1991): 42B08, 42C14

Keyword(s): $N$-fold Fourier integrals, elliptic polynomials, continuous functions from the Sobolev spaces, uniformly convergence

Received May 28, 2009, and in revised form April 13, 2010. (Registered under 72/2009.)