A weak generalized localization criterion for multiple fourier series whose rectangular partial sums are considered over a subsequence

The behavior on measurable set of partial sum as, depending on the smoothness properties of the function, the structural and geometric characteristics of the measurable set, and the constraints imposed on the components of the vector was studied. A weak generalized localization criterion for multiple Fourier series whose rectangular partial sums are considered over a subsequence. The WGL criterion for multiple Fourier series whose rectangular partial sums have index in which precisely component are lacunary. Several theorems are provided to satisfy and prove this set.