Structural and geometric characteristics of sets of convergence and divergence of multiple Fourier series with J k -lacunary sequence of rectangular partial sums

Abstract

We study multiple trigonometric Fourier series of functions f in the classes \(L_p \left( {\mathbb{T}^N } \right)\), p > 1, which equal zero on some set \(\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0\) (µ is the Lebesgue measure), \(\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N\), N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series Sn(x; f) have index n = (n1, ..., nN) ∈ ℤN, in which k (k ≥ 1) components on the places {j1, ..., jk} = Jk ⊂ {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with Jk-lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” Jk) of lacunary sequences in the index n with the structural and geometric characteristics of the set \(\mathfrak{A}\), which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure \(\mathfrak{A}_1\) of the set \(\mathfrak{A}\).