A Weak Generalize Localization of Multiple Fourier Series of Continuous Functions with a Certain Module of Continuity

Let $E$ be an arbitrary measurable set, $E\subset T^N=[-\pi,\pi)^N$, $N\ge 1$, $\mu E>0$, let $\mu$ be a measure. In this paper, a weak generalize almost everywhere localization is studied, i.e., for given subsets $E_1\subset E$, $\mu E_1>0$ we study the almost everywhere convergence of multiple trigonometric Fourier series of functions those are zero on $E$. We obtain sufficient conditions for the almost everywhere convergence of multiple Fourier series (summable over rectangles) of functions from $H^\omega(T^N)$, $\omega(\delta)=o\left(\left[\log\dfrac1\delta\log\log\log\dfrac1\delta\right]^{-1}\right)$, as $\delta\to0$ on $E_1$. These conditions are given in terms of the sets' $E_1$, $E$ structure and geometry and related to certain orthogonal projections of the sets; they are called the $\mathbb{B}_3$ property of the set $E$. Formerly, one of the authors has introduced the $\mathbb B_k$, $k=1,2$ properties of the set $E$, which are related to one-dimensional and two-dimensional projections of the sets $E$ and $E_1$ respectively, as sufficient conditions for the almost everywhere convergence of Fourier series of functions from $L_1(T^N)$ and $L_p(T^N)$, $p>1$. The presented results generalize these ideas.