# Proof of the classical div-curl-lemma

let $1 = \frac{1}{p} + \frac{1}{q}$ as usual. Let $f \in L^p, g \in L^q$ be vector fields from $\mathbb R^n$ to itself.

Assume $div f = 0$ and there exists a function $G$ s.t. $\nabla G = g$. Then $f \cdot g \in \mathcal H^1$ is a Hardy space function.

Do you know where I can find a proof of this conclusion? I am aware of a paper by Coifman et al. "Compensated compactness and Hardy spaces", but I am not granted access to this journal. Hence I am looking for an alternative resource.

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Do you have any sort of library available, or are looking for an online reference? What books have you tried? I bet one of Evans' "Partial Differential Equations," Grafakos' "Classical Fourier Analysis," or Stein's "Harmonic Analysis" have a proof. –  JavaMan Mar 29 '11 at 15:49
Hi @Martin, did you end up looking through those books? –  Glen Wheeler Apr 14 '11 at 7:35
you find it in the book "weak convergence methods...." by evans –  user34172 Jun 21 '12 at 14:09
arxiv.org/abs/0712.2133 may be relevant. –  Willie Wong Jun 21 '12 at 16:08

A remark on the div-curl lemma by P. G. Lemarié-Rieusset gives a complete proof of the result in greater generality, with $L^p-L^q$ replaced by a Calderón-Zygmund pair of Banach spaces.