Non-improvability of sharp endpoint estimates

Zdeněk Mihula, Luboš Pick, Armin Schikorra

For an integer $n$ and the parameter $\gamma \in (0,n)$, the Riesz potential $I_{\gamma }$ is known to take boundedly $L^1({\mathbb{R}}^n)$ into $L^{\frac{n}{n-\gamma },\infty }({\mathbb{R}}^n)$, and also t…