We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function.
Can we find a maximal operator that is integrable for nonzero functions?
More precisely, I would really appreciate some help with the following:
Let $ \phi \in C^\alpha (\mathbb R^n) \cap L^1 (\mathbb R^n) $ and set $ \displaystyle M_\phi f(x) := \sup_{t>0} f * \phi_t $, where $\displaystyle \phi_t (\cdot)= \frac{1}{t^n} \phi \left( \frac{ \cdot} {t} \right)$.
Show that while $\displaystyle |M_\phi| \lesssim Mf $ there exist nonzero function $f \in L^1(R^n)$ such that $ M_\phi f \in L^1 $
Any ideas?
Thanking in advance!