On conditions for the weighted integrability of the sum of the series with monotonic coefficients with respect to the multiplicative systems

In this paper, we studied the issues of integrability with the weight of the sum of series with respect to multiplicative systems, provided that the coefficients of the series are monotonic. The conditions for the weight function and the series' coefficients are found; the sum of the series belongs to the weighted Lebesgue space L-p (1 < p < infinity). In addition, the case of p = 1 was considered. In this case, other conditions for the weighted integrability of the sum of the series under consideration are found. In the case of the generating sequence's boundedness, the proved theorems imply an analogue of the well-known Hardy-Littlewood theorem on trigonometric series with monotone coefficients.