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 Sn(x;f)Sn(x;f) have indices n=(n1,…,nN)∈ZNn=(n1,…,nN)∈ZN (N≥3N≥3), where either N   or N−1N−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 A⊂IN={x∈RN:0≤xj<1,j=1,2,…,N}A⊂IN={x∈RN:0≤xj<1,j=1,2,…,N}, in the case when the structure and geometry of AA are defined by the properties BkBk, 2≤k≤N2≤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 Ω⊂INΩ⊂IN under the minimal conditions imposed on the smoothness of the function on this set.