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

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

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