Sharp weak bounds and limiting weak-type behavior for Hardy type operators

In this paper, Hardy type operator $H_{\beta}$ on $\bR^{n}$ and its adjoint operator $H_{\beta}^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_{\beta}$ and $H_{\beta}^{*}$ being bounded from $L^{p}(|x|^{\alpha})$ to $L^{q,\infty}(|x|^{\gamma})$, and the bounds of the operators $H_{\beta}$ and $H_{\beta}^{*}$ are sharp worked out. In particular, when $\alpha=\gamma=0$, the norm of $H_{\beta}$ is equal to $1$. The other is that we study limiting weak-type behavior for the operator $H_{\beta}$ and its optimal form was obtained.