Necessary conditions for L1-convergence of double Fourier series

We extend the results of A.S. Belov from single to double Fourier series, which give necessary conditions in terms of the Fourier coefficients for L1-convergence. Our basic tools are Hardy's inequality for the Taylor coefficients of a function in the Hardy space H1 on the unit disk, and the Bernstein–Zygmund inequalities for the derivative of a trigonometric polynomial in L1-norm.