# Poincaré inequality for curl-integrable functions

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 $d=2$.

• I'm not sure that it will contain the answer, but the following paper treats a very similar subject: Note on explicit proof of Poincare inequality for differential forms on manifolds by Leonid Shartser (arXiv:1010.3356). May 29, 2015 at 17:30
• Looks interesting. But figuring out whether this gives what I want seems a non-trivial task with my geometry skills. I was hoping for a more definite answer for the question. Jun 2, 2015 at 7:48
• Do you mean Friedrich's inequality? Sep 23, 2015 at 14:30
• To best of my knowledge Friedrich's inequality requires that the function is in some Sobolev space (i.e. all the partial derivatives, up to some order, are integrable). I was hoping to obtain something for merely curl-integrable functions (say, in 2D). Oct 2, 2015 at 8:12

Counterexample $(x,-y)$, or am I misunderstanding something here.