Summability in Musielak--Orlicz Hardy spaces
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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.
Abstract
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.
참고문헌 (31)
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