Summability in Anisotropic Musielak–Orlicz Hardy Spaces

Let φ Rⁿ × [0,∞) → [0,∞) be an anisotropic growth function and A a general expansive matrix on Rⁿ. Let HAφ(Rⁿ) be the anisotropic Musielak–Orlicz Hardy space associated with A. In this paper, a general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms in HAφ(Rⁿ). Precisely, the author establishes the boundedness of maximal operators, induced by the so-called θ-means, from HAφ(Rⁿ) to the Musielak–Orlicz space Lφ(Rⁿ). As applications, some norm and almost everywhere convergence results of the θ-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.