TITLE: Poincaré inequality for curl-integrable functions
QUESTION [6 upvotes]: 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 $d=2$.

REPLY [2 votes]: Counterexample $(x,-y)$, or am I misunderstanding something here.