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SUMMABILITY IN MUSIELAK-ORLICZ HARDY SPACES
Jun Liu,Haonan Xia Korean Mathematical Society 2023 대한수학회지 Vol.60 No.5
Let 𝜑 : ℝ<sup>n</sup> × [0, ∞) → [0, ∞) be a growth function and H<sup>𝜑</sup>(ℝ<sup>n</sup>) the Musielak-Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called 𝜃-summability is considered for multi-dimensional Fourier transforms in H<sup>𝜑</sup>(ℝ<sup>n</sup>). Precisely, with some assumptions on 𝜃, the authors first prove that the maximal operator of the 𝜃-means is bounded from H<sup>𝜑</sup>(ℝ<sup>n</sup>) to L<sup>𝜑</sup>(ℝ<sup>n</sup>). As consequences, some norm and almost everywhere convergence results of the 𝜃-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner-Riesz, Weierstrass and Picard-Bessel summations, are also presented.
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