Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$

The concept of weak generalized localization almost everywhere is introduced. For the multiple Fourier series of a function $f$, weak generalized localization almost everywhere holds on the set $E$ ($E$ is an arbitrary set of positive measure $E\subset T^N=[-\pi,\pi]^N$) if the condition $f(x)\in L_p(T^N)$, $p\geqslant1$, $f=0$ on $E$ implies that the indicated series converges almost everywhere on some subset $E_1\subset E$ of positive measure. For a large class of sets $\{E\}$, $E\subset T^N$, a number of propositions are proved showing that weak localization of rectangular sums holds on the set $E$ in the classes $L_p$, $p\geqslant1$, if and only if the set $E$ has certain specific properties. In the course of the proof the precise geometry and structure of the subset $E_1$ of $E$ on which the multiple Fourier series converges almost everywhere to zero are determined.
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