Optimal weak estimates for Riesz potentials

TL;DR

This paper establishes a sharp reverse weak estimate for Riesz potentials in Lebesgue spaces, determines the explicit constant, and analyzes its asymptotic behavior as the parameter approaches zero.

Contribution

It provides the first sharp reverse weak estimate for Riesz potentials and investigates the asymptotic behavior of the best constant in the estimate.

Findings

Established a sharp reverse weak estimate with explicit constant.

Proved the asymptotic behavior of the best constant as s approaches zero.

Analyzed the dependence of the constant on parameters for Riesz potentials.

Abstract

In this note we prove a sharp reverse weak estimate for Riesz potentials $$\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\geq \gamma_sv_{n}^{\frac{n-s}{n}}\|f\|_{L^1}~~\text{for}~~0<f\in {L^1(\mathbb{R}^n)},$$ where $\gamma_s=2^{-s}\pi^{-\frac{n}{2}}\frac{\Gamma(\frac{n-s}{2})}{\Gamma(\frac{s}{2})}$. We also consider the behavior of the best constant $\mathcal{C}_{n,s}$ of weak type estimate for Riesz potentials, and we prove $\mathcal{C}_{n,s}=O(\frac{\gamma_s}{s})$ as $s\rightarrow 0$.