Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

I. L. Bloshanskii, S. K. Bloshanskaya, D. A. Grafov

Published 2017-04-15Version 1

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\subset M$ the Weyl multiplier for convergence of these series has the form $W(\nu)=\prod \limits_{j=1}^{N-k} \log(|\nu_{{\alpha}_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu=(\nu_1,\dots,\nu_N)\in{\mathbb Z}^N$. So, the "one-dimensional" Weyl multiplier -- $\log(|\cdot|+2)$ -- presents in $W(\nu)$ only on the places of "free" (nonlacunary) components of the vector $\nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more "free" components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.