$u$-convergence of multiple Fourier series

The $u$-convergence of multiple Fourier series is studied, generalizing convergence in the Pringsheim sense, with respect to spheres. A definitive condition in terms of moduli of smoothness is found on a functional class that implies the $u$-convergence of Fourier series in the metrics $L_p(T^m)$, where $1\leqslant p\leqslant\infty$, $p\ne 2$ and $m\geqslant 2$.