Is the maximal function bounded on the Besov space? The Hardy-Littlewood maximal function of a function $f$ is defined by
$$
M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy,
$$
where $|B_r|$ denotes the Lebesgue measure of the ball $B_r$. It is well-known that for $p\in(1,\infty]$, there exists a constant $C_{d,p}>0$ such that
\begin{align}
\|M f\|_p\leq C_{d,p}\|f\|_p.   \label{mf}
\end{align}
That is, $M$ is bounded in $L^p$-spaces. 
The zero order Besov space can be defined as follows. Recall the Littlewood-Paley operators is deifned as
$$
\Lambda_jf:=h_j*f \quad (convolution),
$$
where $h_j\in C_0^\infty(R^n)$.
We have
$$
f=\sum_{j\geq -1}\Lambda_jf.
$$
Then, $B^0_{p,\infty}(R^n)$ is defined as $f$ satisfying
$$
\sup_{j}\|\Lambda_jf\|_p<\infty.
$$
Is the operator $M$ bounded in $B^0_{p,\infty}(R^n)$? (the problem I met is the change of order for $M$ and convolution. Moreprecisely, is it true that for a $C>0$
$$
\Lambda_jMf\leq CM\Lambda_jf??
$$
If this is true, then $\|\Lambda_jMf\|_p \leq C\|M\Lambda_jf\|_p\leq C\|\Lambda_jf\|_p$, this means that $\|Mf\|_{B^0_{p,\infty}}\leq C_d\|f\|_{B^0_{p,\infty}}$.)
 A: *

*The original inequality in the question was  $\Lambda_jMf\leq M\Lambda_j f$.
This inequality is not true in general even if $f\geq 0$ since the opposite inequality is true.


Assume that $h_j$ and $f$ are non-negative. Let $M_r=|B(0,r)|^{-1}\chi_{B(0,r)}$. Then $Mf=\sup_{r>0} M_r*f$. Hence
$$
h_j*M_r*f=M_r*h_j*f,
\qquad
h_j*Mf\geq M_r*h_j*f,
$$
because $M_rf\leq Mf$ and taking the supremum over $r$ yields
$$
\Lambda_jMf= h_j*Mf\geq M(h_j*f)=M\Lambda_j f.
$$
If your inequality were true $\Lambda_jMf\leq M\Lambda_j f$, then we would have equality: $\Lambda_jMf= M\Lambda_j f$ which holds rarely. I do not have an example on top of my head, but such an example is easy to find.


*

*If we drop the assumption that $f\geq 0$, then it is easy to show that even a stronger inequality from the current formulation of the problem: $\Lambda_jMf\leq CM\Lambda_j f$ is not true. Here is an example.


Let $R>0$ be large and let $\lambda_j\approx \chi_{B(0,R)}$ be a $C_0^\infty$ approximation of the characteristic function of the ball. Let $f\in C_0^\infty(B(0,1))$, $\int f=0$, $\int |f|=1$. Then 
$|\Lambda_j f|=|\lambda_j*f|\leq 1$
and $\Lambda_j f\neq 0$ in a neighborhood of $\partial B(0,R)$ of thickness $1$. 
For $|x|$ much larger than $R$ we have:
$$
M\Lambda_j f\approx \frac{R^{n-1}}{|x|^n}.
$$
However,
$$
M f\approx \frac{1}{|x|^n}
\qquad
\text{so}
\qquad
\Lambda_jMf(x)\approx \frac{R^n}{|x|^n}
$$
and the inequality $\Lambda_jMf\leq CM\Lambda_j f$ would require $C$ to be of magnitute $R$.


*

*One easily can generalize the previous example to the case in which $\lambda_j$ is supported in an annulus which, I believe, is the case you are interested in.

