Two-dimensional Waterman classes and $u$-convergence of Fourier series

New results on the $u$-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than $BV(T^2)$ ensures even the uniform boundedness of the $u$-sums of the double Fourier series of functions in this class. On the other hand, the concept of $u(K)$-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions $f(x,y)$ belonging to the class $\Lambda_{1/2}BV(T^2)$, where $\Lambda_a=\biggl\{\dfrac{n^{1/2}}{{(\ln(n+1))}^a}\biggr\}_{n=1}^\infty$, the corresponding $u(K)$-partial sums are uniformly bounded, while if $f(x,y)\in\Lambda_aBV(T^2)$, where $a<\frac12$, then the double Fourier series of $f(x,y)$ is $u(K)$-convergent everywhere.