Weight Inequalities for the Sum of Series With Respect to the Multiplicative Systems

This paper investigates series over Price multiplicative systems with coefficients belonging to the class of sequences of bounded variation. Conditions are obtained for estimating the norm of the sum of such series in weighted Lebesgue spaces. These conditions are formulated in terms of the weight function and the corresponding weight sequence. The methodology relies on techniques of harmonic analysis, the Abel transformation, and the Muckenhoupt criteria for the boundedness of the Hardy operator in weighted Lebesgue spaces. Additionally, discrete three-weight Hardy inequalities are considered, and their applicability to the analyzed series is examined. The main theorems establish a relationship between the variation of the coefficients and the integral characteristics of the weights. The results extend the applicability of known analytical methods to a wider class of functional series and are of interest in harmonic analysis, series theory, and the estimation of solutions to differential equations in functional spaces.