Almost everywhere convergence over cubes of multiple trigonometric Fourier series

Under certain conditions on a function $\varphi\colon[0,+\infty)\to[0,+\infty)$ we prove a theorem asserting that the convergence almost everywhere of trigonometric Fourier series for all functions of class $\varphi(L)_{[-\pi,\pi)}$ implies the convergence over cubes of the multiple Fourier series and all its conjugates for an arbitrary function $f\in\varphi(L)(\log^+L)^{d-1}_{[-\pi,\pi)^d}$, $d\in\mathbb N$. It follows from this and an earlier result of the author on the convergence almost everywhere of Fourier series of functions of one variable and class $L(\log^+L)(\log^+\log^+\log^+L)_{[-\pi,\pi)}$ that if $f\in L(\log^+L)^d(\log^+\log^+\log^+L)_{[-\pi,\pi)^d}$, $d\in\mathbb N$, then the Fourier series of $f$ and all its conjugates converge over cubes almost everywhere.