Summability in anisotropic mixed-norm Hardy spaces

Let $H_A^{\overrightarrow{p}}({\mathbb{R}}^n)$ be the anisotropic mixed-norm Hardy space, where $\overrightarrow{p}\in (0,\mathrm{\infty }{)}^n$ 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 $H_A^{\overrightarrow{p}}({\mathbb{R}}^n)$. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $\theta$-means, from $H_A^{\overrightarrow{p}}({\mathbb{R}}^n)$ to the mixed-norm Lebesgue space $L^{\overrightarrow{p}}({\mathbb{R}}^n)$. 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.