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!