We discuss the boundedness of the cone-like restricted maximal operator of two-dimensional Nörlund means from the dyadic Hardy space \(H^\gamma _p({\mathbb I}^2)\) to the Lebesgue space \(L_p({\mathbb I}^2)\) . The results are formulated for monotone non-increasing and monotone non-decreasing generating sequences of the Nörlund means. We give a necessary and sufficient condition for the boundedness of the cone-like restricted maximal operator of two-dimensional Nörlund means from the dyadic Hardy space \(H^\gamma _p({\mathbb I}^2)\) to the Lebesgue space \(L_p({\mathbb I}^2)\) ( \(1/2<p\leqslant 1\) ), when both of the generating sequences are monotone non-increasing. The almost everywhere convergence of the two-dimensional Nörlund means is proved under certain assumptions in the case when its indices belong to some positive cone-like set around the function \(\gamma \) .
