We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the -convergence of double Fourier series. We also give necessary and sufficient conditions for the -convergence under appropriate assumptions.
We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the -metric, where . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the -metric, where .