Summability in Anisotropic Musielak–Orlicz Hardy Spaces

Abstract

Let $\varphi \colon \mathbb{R}^{n} \times [0,\infty) \to [0,\infty)$ be an anisotropic growth function and $A$ a general expansive matrix on $\mathbb{R}^{n}$. Let $H_{A}^{\varphi}(\mathbb{R}^{n})$ be the anisotropic Musielak–Orlicz Hardy space associated with $A$. In this paper, a general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H_{A}^{\varphi}(\mathbb{R}^{n})$. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $\theta$-means, from $H_{A}^{\varphi}(\mathbb{R}^{n})$ to the Musielak–Orlicz space $L^{\varphi}(\mathbb{R}^{n})$. As applications, some norm and almost everywhere convergence results of the $\theta$-means, which generalize the well-known Lebesgue's theorem, are presented. Finally, the corresponding conclusions of two well-known specific summability methods, that is, Bochner–Riesz and Weierstrass means, are also obtained.