Generalized localization for the double trigonometric Fourier series and the Walsh–Fourier series of functions in $L\log^+L\log^+\log^+L$

For an arbitrary open set $\Omega\subset I^2=[0,1)^2$ and an arbitrary function $f\in L\log^+L\log^+\log^+L(I^2)$ such that $f=0$ on $\Omega$ the double Fourier series of $f$ with respect to the trigonometric system $\Psi=\mathscr E$ and the Walsh–Paley system $\Psi=W$ is shown to converge to zero (over rectangles) almost everywhere on $\Omega$. Thus, it is proved that generalized localization almost everywhere holds on arbitrary open subsets of the square $I^2$ for the double trigonometric Fourier series and the Walsh–Fourier series of functions in the class $L\log^+L\log^+\log^+L$ (in the case of summation over rectangles). It is also established that such localization breaks down on arbitrary sets that are not dense in $I^2$, in the classes $\Phi_\Psi(L)(I^2)$ for the orthonormal system $\Psi=\mathscr E$ and an arbitrary function such that $\Phi_{\mathscr E}(u)=o(u\log^+\log^+u)$ as $u\to\infty$ or for $\Phi_W(u)=u(\log^+\log^+u)^{1-\varepsilon}$, $0<\varepsilon<1$.