Weak type estimates for Bochner--Riesz means on Hardy-type spaces associated with ball quasi-Banach function spaces

Let $X\left(\mathbb{R}^{n}\right)$ be a ball quasi-Banach function space on $\mathbb{R}^{n}$, $WX\left(\mathbb{R}^{n}\right)$ be the weak ball quasi-Banach function space on $\mathbb{R}^{n}$, $H_{X}\left(\mathbb{R}^{n}\right)$ be the Hardy space associated with $X\left(\mathbb{R}^{n}\right)$ and $WH_{X}\left(\mathbb{R}^{n}\right)$ be the weak Hardy space associated with $X\left(\mathbb{R}^{n}\right)$. In this paper, we obtain the boundedness of the Bochner--Riesz means and the maximal Bochner--Riesz means from $H_{X}\left(\mathbb{R}^{n}\right)$ to $WH_{X}\left(\mathbb{R}^{n}\right)$ or $WX\left(\mathbb{R}^{n}\right)$, which includes the critical case. Moreover, we apply these results to several examples of ball quasi-Banach function spaces, namely, weighted Lebesgue spaces, Herz spaces, Lorentz spaces, variable Lebesgue spaces and Morrey spaces. This shows that all the results obtained in this article are of wide applications, and more applications of these results are predictable.