In $L^p(\mathbb{R}^3)$, it holds for $1< p< \infty$ and $\mu\in C^\infty_0(\mathbb{R}^3)$, $$\|\nabla\mu\|_p\leq C \left( \|\operatorname{div} \mu\|_p + \|\nabla\times\mu\|_p \right).$$ So, how about in $\bar{M}^p(\mathbb{R}^3)$, here \begin{equation*} \|\mu\|_{\bar{M}^p}=\sup_{x\in\mathbb{R}^3,r>0}r^{\frac{-3}{p'}}|\mu|(B(x,r))<\infty,\,p'=\frac{p}{p-1} \end{equation*} and $|\mu|$ is the total variation of $\mu$.
Is it also right for $1< p< \infty$, $$\|\nabla\mu\|_{\bar{M}^p}\leq C \left( \|\operatorname{div} \mu \|_{\bar{M}^p} + \|\nabla\times\mu\|_{\bar{M}^p}\right).$$
