Optimal weak estimates 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 $s\rightarrow 0$.