Restricted convergence of two-dimensional Nörlund means of Walsh–Fourier series

Source abstract

Abstract We discuss the boundedness of the cone-like restricted maximal operator of two-dimensional Nörlund means from the dyadic Hardy space $$H^\gamma _p({\mathbb I}^2)$$ H p γ ( I 2 ) to the Lebesgue space $$L_p({\mathbb I}^2)$$ L p ( 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 $$H^\gamma _p({\mathbb I}^2)$$ H p γ ( I 2 ) to the Lebesgue space $$L_p({\mathbb I}^2)$$ L p ( I 2 ) ( $$1/2<p\leqslant 1$$ 1 / 2 &lt; p ⩽ 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 $$\gamma$$ γ .