Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and $$ u_B := \frac1{|B|}\int_B u \, dx. $$ The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, then $$ \|u-u_B\|_{L^\kappa(B)} \le Cr \|\nabla u\|_{L^p(B)} $$ for $$ \kappa = \frac{dp}{d-p}, \quad 1 < p < d. $$
My question is whether something similar is true for merely curl-integrable functions. More precisely, suppose $u \in L_{\rm loc}^p$ is divergence-free and $\text{curl } u \in L_{\rm loc}^p, p>1$. As pointed out below, a standard Poincaré inequality with the gradient replaced by the curl, $$ \|u-u_B\|_{L^\kappa(B)} \le Cr \|\text{curl }u\|_{L^p(B)}, $$ cannot hold in general. But I suspect there should be a corresponding result with some extra assumptions or with some additional term on the right hand side. Any references in to that direction are appreciated.
I am mostly interested in the case $d=2$ and $1<p<2$.
Edit: If it is easier, I would also be interested just in the case, where $\kappa$ is replaced by $p \in (1,2)$ for $d=2$.
