Geodesic
Weighted integrability of sums of series with respect to multiplicative systems
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 129-136

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A necessary and sufficient condition for $L^p$-integrability with power weight of a function $f$ represented by the series with respect to multiplicative systems with generalized monotone coefficients is obtained. The integrability of the majorant of partial sums of a representing series is also described by the same conditions. In addition we study the integrability of difference quotient $(f(x)-f(0))/x$. 
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     author = {S. S. Volosivets and R. N. Fadeev},
     title = {Weighted integrability of sums of series with respect to multiplicative systems},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {129--136},
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S. S. Volosivets; R. N. Fadeev. Weighted integrability of sums of series with respect to multiplicative systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 129-136. http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a1/