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

TL;DR

This paper establishes sharp bounds and analyzes the limiting weak-type behavior of Hardy type operators on weighted Lebesgue spaces, providing precise operator norms and insights into their asymptotic properties.

Contribution

It introduces novel methods to determine sharp bounds for Hardy type operators and characterizes their limiting weak-type behavior, including the exact operator norm when weights are trivial.

Findings

Operators are bounded from weighted L^p to weak L^q spaces with sharp bounds.

The norm of H_β is exactly 1 when weights are trivial.

Optimal limiting weak-type behavior of H_β is characterized.

Abstract

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.