On $\Lambda$-convergence almost everywhere of multiple trigonometric Fourier series

We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the $\lambda$-convergence for $\lambda >1$. The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class $L (\ln ^ + L) ^ d \ln ^ + \ln ^ + \ln ^ + L ([0,2 \pi)^d )$ has been generalized to the case of the $\Lambda$-convergence for some sequences $\Lambda$.