ACTA issues

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

Acta Sci. Math. (Szeged) 83:3-4(2017), 511-537

Abstract. 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 $n=(n_1,\dots,n_N) \in{\msbm Z}^N$, $N\ge3$, 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 $W(\nu )=\prod_{j=1}^{N-k} \log(|\nu_{{\alpha }_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu =(\nu_1,\dots,\nu_N)\in{{\msbm Z}}^N$. 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 Orlicz 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 $N\geq3$ even in the class of continuous functions.

DOI: 10.14232/actasm-017-275-8

AMS Subject Classification (1991): 42B05

Keyword(s): multiple trigonometric Fourier series, convergence almost everywhere, lacunary sequence, Weyl multipliers

Received April 21, 2017. (Registered under 25/2017.)