The generalized localization for multiple Fourier integrals

In this paper we investigate almost-everywhere convergence properties of the Bochner–Riesz means of N-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner–Riesz means s ⩾ ( N − 1 ) ( 1 / p − 1 / 2 ) , then the Bochner–Riesz means of a function f ∈ L p ( R N ) , 1 ⩽ p ⩽ 2 converge to zero almost-everywhere on R N ∖ supp ( f ) .