A weak generalized localization of multiple Fourier series of continuous functions with a certain module of continuity

Abstract

Let E be an arbitrary measurable set, E ⊂ T N = [−π, π)N, N ≥ 1, μE > 0, and let μ be a measure. In this paper, a weak generalized almost everywhere localization is studied, i.e., for given subsets E 1 ⊂ E, μE 1 > 0 we study the almost everywhere convergence of multiple trigonometric Fourier series of functions that are zero on E. We obtain sufficient conditions for the almost everywhere convergence of multiple Fourier series (summable over rectangles) of functions from {ie031-01}, as δ → 0 on E 1. These conditions are given in terms of the structure and geometry of the sets E 1 and E and are related to certain orthogonal projections of the sets; they are called the {ie031-02} property of the set E. Previously, one of the authors had introduced the {ie031-03}, 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 results presented generalize these ideas.