The "local" Hardy-Littlewood maximal function is given by $$(M\_\phi f)(x)= \sup\_{0<\epsilon<1}|\phi\_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$(Mf)(x)= \sup\_{\epsilon>0}|(\chi\_B)\_\epsilon \ast f|(x),$$
where $\phi\in S(\mathbb{R}^n)$ is some nonnegative, radial, radially decreasing Schwartz function and $\phi\_\epsilon(x)=\epsilon^{-n}\phi(x/\epsilon)$.


The word "local" means that the supremum is only taken on small $\epsilon\in(0,1)$. It seems that this maximal function should have better properties than the classical one. For example, it is well known that if $f\in L^1$, then $Mf\notin L^1$ in general. However, we can easily check that for some $f\in L^1$, we do have $M\_\phi f\in L^1$(e.g. $f(x)=\phi(x)=e^{-|x|^2}$). Are there any references(e.g. books, papers,...) on this "local" maximal function? Thanks!



How much do we know about this "local" Hardy-Littlewood maximal function?
The failure of the maximal function to be a bounded map in $L^1$ is not just an issue of "failure at infinity". 


Consider the case of $n = 1$. Let $f$ be a non-negative function that is compactly supported, and smooth away from the origin. Near the origin suppose that 
$$ f(x) = \frac{1}{|x| (\ln |x|)^2} $$
This function can be easily checked to be in $L^1$, as we can integrate, for $y> 0$ close to zero, 
$$ \int\_0^y f(x) ~\mathrm{d}x = - \frac{1}{\ln y} $$


If you apply your local maximal function to $f$ (with the positivity assumption, WLOG, that $\phi(y) \geq c > 0$ whenever $|y| \leq 1$), you have that for $y\in (0,\epsilon)$ that 
$$ M\_\phi f(y) \gtrsim - \frac{1}{y \ln y} $$
which implies that $M\_\phi f(y)$ is not integrable near the origin. 




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The local properties of the maximal function (the classical one, but it applies almost identically to your version with the Schwartz weight) is remarked upon in Section 5.2, Chapter 1 of Stein's *Singular Integrals and Differentiability Properties of Functions*, where you can find the assertion 
$$ Mf [\log (2 + Mf)]^k \in L^1\_{loc} \iff |f| [\log (2 + |f|)]^{k+1} \in L^1\_{loc} $$
for any non-negative integer $k$. 


