type estimate.

Marcinkiewicz type estimate
---------------------------

In this subsection, we recall the definition of Marcinkiewicz space and prove Marcinkiewicz type estimate.

Let $\Theta\subset {\mathbb{R}}^N$ be a domain and $\mu$ be a positive Borel measure in $\Theta$. For $\kappa>1$, $\kappa'=\kappa/(\kappa-1)$ and $u\in L^1_{loc}(\Theta,d\mu)$, we set $$\label{mod M}
\|u\|_{M^\kappa(\Theta,d\mu)}=\inf\left\{c\in[0,\infty]:\int_E|u|d\mu\le
c\left(\int_Ed\mu\right)^{\frac1{\kappa'}},\ \forall E\subset \Theta,\,E\
\rm{Borel}\right\}$$ and $$\label{spa M}
M^\kappa(\Theta,d\mu)=\{u\in
L_{loc}^1(\Theta,d\mu):\|u\|_{M^\kappa(\Theta,d\mu)}<\infty\}.$$

$M^\kappa(\Theta,d\mu)$ is called the Marcinkiewicz space of exponent $\kappa$, or weak $L^\kappa$-space and $\|.\|_{M^\kappa(\Theta,d\mu)}$ is a quasi-norm.

\[pr 1\] [@BBC] Assume that $1\le q< \kappa<\infty$ and $u\in L^1_{loc}(\Theta,d\mu)$. Then there exists $c_4>0$ dependent of $q,\kappa$ such that $$\int_E |u|^q d\mu\le c_4\|u\|_{M^\kappa(\Theta,d\mu)}\left(\int_E d\mu\right)^{1-q/\kappa},$$ for any Borel set $E$ of $\Theta$.

Now we are ready to state Marcinkiewicz type estimate as follows.

\[general\] Let $\nu\in\mathfrak{M}^b({\mathbb{R}}^N)$, then there exists $c_5>0$ such that $$\label{annex 0}
\|\mathbb{G}_{{\mathbb{R}}^N}[|\nu|]\|_{M^{p^*_\alpha}({\mathbb{R}}^N,d\omega)}\le c_5\|\nu\|_{\mathfrak{M}^b({\mathbb{R}}^N)},$$ where $d\omega(x)=\frac{dx}{1+|x|^{N+2\alpha}}$ and $p^*_\alpha=\frac{N}{N-2\alpha}$.

[*Proof*]{}. For $\lambda>0$ and $y\in {\mathbb{R}}^N$, we set $$A_\lambda(y)=\left\{x\in {\mathbb{R}}^N\setminus\{y\}: G_{{\mathbb{R}}^N}(x,y)>\lambda \right\},\quad
 m_\lambda(y)=\int_{A_\lambda(y)}d\omega.$$ We observe that there exists a positive constant $c_{N,\alpha}$ such that $$G_{{\mathbb{R}}^N}(x,y)= \frac{c_{N,\alpha}}{|x-y|^{N-2\alpha}},\quad (x,y)\in
{\mathbb{R}}^N\times{\mathbb{R}}^N,\ x\neq y,$$ which implies that $$\label{gra 0003}
A_\lambda(y)\subset\left\{x\in {\mathbb{R}}^N: |x-y|\le c_{N,\alpha}\lambda^{-\frac1{N-2\alpha}}\right\}.$$ As a consequence, $$\label{gra 0003wx2}
 m_\lambda(y)=\int_{A_\lambda(y)}\frac{dx}{1+|x|^{N+2\alpha}}\le|A_\lambda(y)| \le c_6\lambda^{-p_\alpha^*},$$ where $c_6>0$ independent of $y$ and $\lambda$ and $p_\alpha^*=\frac{N}{N-2\alpha}$.

Let $E\subset {\mathbb{R}}^N$ be a Borel set and $\lambda>0$, then 