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arXiv · 2608.31043

Sharp constants for weak estimates of the Riesz Potentials

Hanli Tang·Yaojun Wang
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Abstract

In this paper we prove the sharp weak estimates for the Riesz potentials $$\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\leq \gamma_{n,s} v_n^{\frac{n-s}{n}}\frac{\Gamma(s/2)\Gamma((n+2-s)/2)}{\Gamma(n/2)}\|f\|_{L^1},~~ \text{when}~~0<s<\min\{n,2\}$$ and $$ \|I_sf\|_{L^{\frac{n}{n-s},\infty}} \leq \gamma_{n,s}v_n^{\frac{n-s}{n}}\|f\|_{L^1}, ~~\text{when}~~2\leq s<n,$$ 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.

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Hanli Tang, Yaojun Wang. 2026-08-31. Sharp constants for weak estimates of the Riesz Potentials. https://arxiv.org/abs/2608.31043

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