Weak Generalized Localization for Multiple Fourier Series Whose Rectangular Partial Sums Are Considered with Respect to Some Subsequence

In this paper, we obtain the structural and geometric characteristics of some subsets of $\mathbb{T}^N=[-\pi,\pi]^N$ (of positive measure), on which, for the classes $L_p(\mathbb{T}^N)$, $p>1$, where $N\ge 3$, weak generalized localization for multiple trigonometric Fourier series is valid almost everywhere, provided that the rectangular partial sums $S_n(x;f)$  ($x\in\mathbb{T}^N$, $f\in L_p$) of these series have a “number” $n=(n_1,\dots,n_N)\in\mathbb Z_{+}^{N}$ such that some components $n_j$ are elements of lacunary sequences. For $N=3$, similar studies are carried out for generalized localization almost everywhere.