Riesz Potential

Riesz Potential

For $f:{\mathbb{R}}^n\to \mathbb{R}$, $0<\alpha <n$, the Riesz potential of $f$ is defined as $I_{\alpha }f(x)=f\ast |x{|}^{-\alpha }$

Hardy-Littlewood-Sobolev

If $p>1$, $\alpha =n(1-\frac{1}{q}+\frac{1}{p})$, then $\Vert I_{\alpha }f{\Vert }_q\le C(n,p,q)\Vert f{\Vert }_p.$

Corollary

For $S_f(H)=\{x:f(x)>H\}$, with $A\subset {\mathbb{R}}^n,|A|=r^n$, $|S_{I_{\alpha }{\chi }_A}(H)|\le C|S_{I_{\alpha }{\chi }_{B_r}(H)}|$