Optimal weak estimates for Riesz potentials

Abstract

In this note we prove a sharp reverse weak estimate for Riesz potentials $$\Vert I_s(f){\Vert }_{L^{\frac{n}{n-s},\infty }}\ge {\gamma }_s{v}_n^{\frac{n-s}{n}}\Vert f{\Vert }_{L^1}\text{for}0<f\in L^1({\mathbb{R}}^n),$$ 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 $s\to 0$.