Sharp constants for weak estimates of the Riesz Potentials

Abstract

In this paper we prove the sharp weak estimates for the Riesz potentials $$\Vert I_s(f){\Vert }_{L^{\frac{n}{n-s},\infty }}\le {\gamma }_{n,s}{v}_n^{\frac{n-s}{n}}\frac{\Gamma (s/2)\Gamma ((n+2-s)/2)}{\Gamma (n/2)}\Vert f{\Vert }_{L^1},\text{when}0<s<\min \{n,2\}$$ and $$\Vert I_{s}f{\Vert }_{L^{\frac{n}{n-s},\infty }}\le {\gamma }_{n,s}{v}_n^{\frac{n-s}{n}}\Vert f{\Vert }_{L^1},\text{when}2\le s<n,$$ where ${\gamma }_{n,s}=2^{-s}{\pi }^{-\frac{n}{2}}\frac{\Gamma (\frac{n-s}{2})}{\Gamma (\frac{s}{2})}$ and $v_n$ is the volume of the unit ball. The isoperimetric inequality for the Riesz capacity and the Newtonian isocapacitary inequality play a crucial role in our approach.

We obtained the sharp constants for all 0<s<n; Yaojun Wang is now named author