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

I. L. Bloshanskii

Matematičeskie zametki, Tome 71 (2002) no. 4, pp. 508-521

Résumé

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 $L_1(\mathbb T^N)$, $N\ge 2$.