Optimal weak estimates for Riesz potentials

Huang, Liang; Tang, Hanli

Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1123-1131

In this note we prove a sharp reverse weak estimate for Riesz potentials

$$\parallel I_s{\left(f\right)\parallel }_{L^{\frac{n}{n-s},\infty }}\ge {\gamma }_s{v}_n^{\frac{n-s}{n}}{\parallel f\parallel }_{L^1}\text{for}0<f\in L^1\left({ℝ}^n\right),$$

where ${\gamma }_s=2^{-s}{\pi }^{-\frac{n}{2}}\frac{\Gamma \left(\frac{n-s}{2}\right)}{\Gamma \left(\frac{s}{2}\right)}$. We also consider the behavior of the best constant ${𝒞}_{n,s}$ of weak type estimate for Riesz potentials, and we prove ${𝒞}_{n,s}=O\left(\frac{{\gamma }_s}{s}\right)$ as $s\to 0$.

Classification : 42B20
Keywords: Riesz potentials, sharp constant, optimal estimate

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