On the weighted integrability of Price-system series with general monotone and bounded variation coefficients
Gulmira Kenzhebekova, Manshuk Mukhambetzhan, Mendybay Turgumbayev
Source abstract
In this paper, we study series with respect to the Price multiplicative system whose coefficients belong either to the class of general monotone sequences or to the class of sequences of bounded variation. For general monotone coefficients, we obtain an estimate for the remainder of the series. This estimate implies convergence at every point of (0, 1) and uniform convergence on every interval [ε, 1 − ε], 0 < ε < ½. For coefficients of bounded variation, we establish sufficient conditions for the weighted integrability of the sums of such series, together with the corresponding estimates in weighted Lebesgue spaces. We also obtain necessary conditions, expressed in terms of suprema over a special sequence of indices (mₙ). The main tool is a discrete three-weight Hardy inequality on the cone of nonnegative non-increasing sequences. These results extend earlier work on series with monotone coefficients and with coefficients of bounded variation.
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