$U$-convergence almost everywhere of double Fourier series

In this paper we consider $u$-convergence of double Fourier series ($u$-convergence of a double number series implies that it converges in the sense of Pringsheim, over spheres, and so on.) Unextendable classes of Weyl multipliers for $u$-convergence almost everywhere are described. In addition, close to exact sufficient conditions for $u$-convergence almost everywhere in the spaces $L(T^2)$ and $L_2(T^2)$ are found.