Journal of the
Korean Mathematical Society JKMS

Let $\varphi: \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be a growth function and $H^{\varphi}(\mathbb{R}^n)$ the Musielak--Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H^{\varphi}(\mathbb{R}^n)$. Precisely, with some assumptions on $\theta$, the authors first prove that the maximal operator of the $\theta$-means is bounded from $H^{\varphi}(\mathbb{R}^n)$ to $L^{\varphi}(\mathbb{R}^n)$. As consequences, some norm and almost everywhere convergence results of the $\theta$-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.

Keywords: Musielak--Orlicz Hardy space, summability, Bochner--Riesz summation, Weierstrass summation, maximal operator

MSC numbers: Primary 42B35, 42B30, 42B08

Supported by: This project was financially supported by the National Natural Science Foundation of China (Grant No.~12001527), the Natural Science Foundation of Jiangsu Province (Grant No.~BK20200647) and the Postdoctoral Science Foundation of China (Grant No.~2021M693422).