Some Estimates for Maximal Bochner-Riesz Means on Musielak-Orlicz Hardy Spaces

Let $\varphi\colon\mathbb{R}^n\times[0,\infty) \to [0,\infty)$ satisfy that $\varphi(x,\,\cdot\,)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $\varphi(\,\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t\in(0,\infty)$. The Musielak–Orlicz Hardy space $H^\varphi(\mathbb{R}^n)$ is defined to be the space of all tempered distributions whose grand maximal functions belong to the Musielak–Orlicz space $L^\varphi(\mathbb{R}^n)$. In this paper, the authors establish the boundedness of maximal Bochner–Riesz means $T^\delta_\ast$ from $H^\varphi(\mathbb{R}^n)$ to $WL^\varphi(\mathbb{R}^n)$ or $L^\varphi(\mathbb{R}^n)$. These results are also new even when $\varphi(x,t):=\Phi(t)$ for all $(x,t)\in\mathbb{R}^n\times[0,\infty)$, where $\Phi$ is an Orlicz function.