Sharp constants for weak estimates of Riesz Potentials when $0<s<\min\{n,2\}$

Abstract

In this paper we prove the sharp weak estimate for the Riesz potentials for $0<s<\min\{n,2\}$ $$\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\leq γ_{n,s} v_n^{\frac{n-s}{n}}\frac{Γ(s/2)Γ((n+2-s)/2)}{Γ(n/2)}\|f\|_{L^1}.$$ where $γ_{n,s}=2^{-s}π^{-\frac{n}{2}}\frac{Γ(\frac{n-s}{2})}{Γ(\frac{s}{2})}$. The isoperimetric inequality for the Riesz capacity plays a crucial role in our approach.