TITLE: Approximation of the identity and Hardy-Littlewood maximal function
QUESTION [10 upvotes]: 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.

REPLY [4 votes]: Actually you can start from the most simple case: when $K_\delta$ is approximated by simple function $\phi$. $$\phi=\sum^{m}_{j=1} C_{j}\chi_{B_j},C_j\geq0, \ \  f*\phi=\sum^{m}_{j=1} C_{j} |B_j|\cdot\frac{1}{|B_j|}\cdot f*\chi_{B_j}$$. $$\therefore |f*\phi|\leq \sum^{m}_{j=1} C_{j} |B_j|f^{*}=\int \phi \cdot f^{*}=||\phi||_{1}\cdot f^*$$.
Then use ${\phi_{k}}$to approach $K_\delta.$