A Criterion for Weak Generalized Localization in the Class $L_1$ for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f\in L_p$, $p\ge 1$, $f(x)=0$, on a set of positive measure $\mathfrak A\subset \mathbb T^N=[-\pi ,\pi )^N$, $N\ge 2$, depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal F$, where $\mathcal F$ is the rotation group about the origin in $\mathbb R^N$. This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{-1}({\mathfrak A})\cap \mathbb T^N$ (where $\tau ^{-1}\in \mathcal F$ satisfies $\tau ^{-1}\cdot \tau =1$) and $\mathbb T^N\setminus \operatorname {supp}(f\circ \tau )$ depending on the rotation, i.e., on $\tau \in \mathcal F$. In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f\circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1(\mathbb T^N)$, $N\ge 2$.