Summability in Anisotropic Musielak–Orlicz Hardy Spaces

Let R^n [0, ) [0, ) be an anisotropic growth function and A a general expansive matrix on R^n. Let H₀^ (R^n) be the anisotropic Musielak–Orlicz Hardy space associated with A. In this paper, a general summability method, the so-called -summability is considered for multi-dimensional Fourier transforms in H₀^ (R^n). Precisely, the author establishes the boundedness of maximal operators, induced by the so-called -means, from H₀^ (R^n) to the Musielak–Orlicz space L^ (R^n). As applications, some norm and almost everywhere convergence results of the -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.