L1-Convergence of Double Fourier Series

We prove the convergence in L1-norm of the double Fourier series of an integrable function ƒ(x, y) which is 2π-periodic with respect to x and y, with coefficients ajk satisfying certain conditions of the Hardy-Karamata kind, and such that ajklogj logk→0 as j, k →∞. We consider separately double cosine, sine, cosine-sine, and complex trigonometric series