Weighted integrability of double trigonometric series

We study the double trigonometric series whose coefficients $c_{jk}$ are such that $\sum _{j=-\infty}^\infty\sum _{k=-\infty}^\infty |c_{jk}|<\infty.$ Then its rectangular partial sums converge uniformly to some $f\in C(T^2)$. We give sufficient conditions for the Lebesgue integrability of $\{f(x,y)-f(x,0)-f(0,y)+f(0,0)\}\phi(x,y)$, where $\phi(x,y)=1/xy, 1/x$, or $1/y$. For certain cases, they are also necessary conditions. Our results extend those of Boas and Móricz from the one-dimensional to the two-dimensional series.