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 ) ∈ 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 A ⊂ I N = { x ∈ R N : 0 ≤ x j < 1 , j = 1 , 2 , … , N } , in the case when the structure and geometry of A 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 Ω ⊂ I N under the minimal conditions imposed on the smoothness of the function on this set. [ABSTRACT FROM AUTHOR]