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

In this paper, Hardy type operator $H_{\beta }$ on ${\text{bR}}^n$ and its adjoint operator $H_{\beta }^{\ast }$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_{\beta }$ and $H_{\beta }^{\ast }$ being bounded from $L^p(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ to $L^{q,\mathrm{\infty }}(\mathrm{\mid }x{\mathrm{\mid }}^{\gamma })$, and the bounds of the operators $H_{\beta }$ and $H_{\beta }^{\ast }$ 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.