Weighted integrability of double series with respect to multiplicative systems

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.