Convergence and localization in Orlicz classes for multiple Walsh-Fourier series with a lacunary sequence of rectangular partial sums

For functions f in Orlicz classes, we consider multiple Walsh-Fourier series for which the rectangular partial sums S-n(x; f) have indices n = (n(1), ..., n(N)) is an element of Z(N) (N >= 3), where either N or N-1 components are elements of (single) lacunary sequences. For this series, we prove the validity of weak generalized localization almost everywhere on an arbitrary measurable set U subset of I-N = {x is an element of R-N : 0 <= x(j) < 1, j = 1,2, ..., N}, in the case when the structure and geometry of U are defined by the properties B-k, 2 <= k <= N. We define the relation between the parameter k and the "smoothness" of functions in terms of the Orlicz classes. As a consequence, we obtain some results on the "local smoothness conditions." In particular, the theorem is proved for the convergence of Walsh-Fourier series on an arbitrary open set Omega C I-N under the minimal conditions imposed on the smoothness of the function on this set. (C) 2015 Elsevier Inc. All rights reserved.