Necessary conditions for -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 L 1 -convergence. Our basic tools are Hardyʹs inequality for the Taylor coefficients of a function in the Hardy space H 1 on the unit disk, and the Bernstein–Zygmund inequalities for the derivative of a trigonometric polynomial in L 1 -norm.