Geodesic
Weighted integrability of double series with respect to multiplicative systems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 69-87.

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Necessary and sufficient conditions for $L^p$-integrability with power weight of a function $f$ represented by the double series with respect to a multiplicative system with generalized monotone coefficients are obtained. These conditions are given in terms of the coefficients or their second mixed differences. In addition, the integrability of the difference quotient $(f(x,y)-f(x,0)-f(0,y)+f(0,0))/(xy)$ is studied. 
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S. S. Volosivets; R. N. Fadeev. Weighted integrability of double series with respect to multiplicative systems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 69-87. http://geodesic.mathdoc.fr/item/FPM_2013_18_5_a3/

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