Continuity in $\Lambda$-variation of functions of several variables and convergence of multiple Fourier series

The behaviour of rectangular partial sums of the Fourier series of functions of several variables having bounded $\Lambda$-variation is considered. It is proved that if a continuous function is also continuous in harmonic variation, then its Fourier series uniformly converges in the sense of Pringsheim. On the other hand, it is demonstrated that in dimensions greater than 2 there always exists a continuous function of bounded harmonic variation with Fourier series divergent over cubes at the origin.