Hedberg inequality
= Hedberg inequality
{c}
If $1<p<n/\alpha$, the pointwise Hedberg inequality is
$$
I_\alpha|f|(x)
\leq C_{n,p,\alpha}
\lVert f\rVert_p^{\alpha p/n}
(Mf(x))^{1-\alpha p/n}.
$$
Split the integral at radius $r$. Dyadic annuli bound the near part by $C r^\alpha Mf(x)$, while <Hölder's inequality> bounds the far part by $C\lVert f\rVert_p r^{\alpha-n/p}$. Choosing $r^{n/p}=\lVert f\rVert_p/Mf(x)$ balances the terms.