The authors consider absolutely convergent double trigonometric series \[ \sum_{j\in\mathbb{Z}} \sum_{k\in\mathbb{Z}} c_{jk}e^{i(jx+ky)}=: f(x,y), \] and give sufficient conditions for the Lebesgue integrability of the function \(\{f(x,y)-f(x,0)-f(0,y)+f(0,0)\}\varphi(x,y)\), where \(\varphi(x,y) =1/xy\) or \(1/x\) or \(1/y\). In particular cases, the conditions turn out to be necessary.
These results extend those of R. P. Boas jun. [“Integrability theorems for trigonometric transforms” (1967; Zbl 0145.06804)] and the reviewer [Acta. Sci. Math. 57, No. 1-4, 443-452 (1993; Zbl 0879.42005)] from single to double series.