A weak generalized localization criterion for multiple Walsh-Fourier series with J k -lacunary sequence of rectangular partial sums

We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure , , N a parts per thousand yen 3 (in terms of the structure and geometry of the set ), for multiple Walsh-Fourier series (summed over rectangles) of functions f in the classes , p > 1 (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure of the set , when the function expanded in a series equals zero on ), in the case when the rectangular partial sums S (n) (x; f) of this series have indices n = (n (1), aEuro broken vertical bar, n (N) ) a a"currency sign (N) in which some components are elements of (single) lacunary sequences.