Summability in anisotropic mixed-norm Hardy spaces

Abstract

<abstract><p>Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.</p></abstract>