The inequality seems to be simple but I could not find the right limits of integration.
$\sup_{\delta>0} |f*K_{\delta}|(x)\leq c f^*(x)$
Where is some positive constant, $f$ is integrable, $K_\delta$ is an approximation of the identity and $f^*$ is the Hardy-Littlewood maximal function of $f$.
An approximation of the identity is family of Kernel satisfying:
I)$\int_{\mathbb{R}^n}K_{\delta}(x)dx = 1$;
II)$|K_{\delta}(x)|\leq A\delta^{-n}$;
III)$|K_{\delta}(x)|\leq A\delta /|x|^{n+1}$.
And the maximal function is the non-centered.