Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m geqslant 2$ and a $2pi $ -periodic (in each variable) function $f(x) in C(T^m )$ belongs to the Nikol'skii class $h_infty ^{(m - 1)/2} (T^m )$ , then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_infty ^{(m - 1)/2} (T^m )$ whose Fourier series is divergent over hyperbolic crosses at some point.