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When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states

Let $\Omega \subset \mathbb R^n$ be a Lipschitz domain. If $f \in H^{-1}(\Omega)$ and $\nabla f\in H^{-1}(\Omega)$. Then $f \in L^2(\Omega)$.

I have looked for the proof of this lemma on the Internet and found several links, such as this, this and this. It seems all proofs of the lemma have something to do with Fourier transform. To be more precise, we first prove the special case where $\Omega = \mathbb R^n$ using Fourier transform, then we try to prove the general case using the special case. The full proof is quite tedious, and I think there is a lack of intuition.

I think the lemma itself is quite concise and promising. Maybe we can prove it in a more direct way, which is, not invoking Fourier transform? I think at least it should be possible when $n = 1$, though I don't know how to prove it, either.

Thank you very much if you would like to help.

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    $\begingroup$ Lion's lemma is equivalent to several other properties that have simpler proofs, see On a lemma of Jacques-Louis Lions and its relation to other fundamental results $\endgroup$
    – Carlo Beenakker
    Commented Jan 13 at 11:33
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    $\begingroup$ @CarloBeenakker I think you should post it as an answer. The paper you quote provides a very nice answer to the question and a very useful reference. Posting the comment as an answer will increase its visibility and many people might find the reference useful. $\endgroup$
    – Piotr Hajlasz
    Commented Jan 13 at 13:17
  • $\begingroup$ thank you, @PiotrHajlasz , for the encouragement; done. $\endgroup$
    – Carlo Beenakker
    Commented Jan 13 at 13:35

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Lion's lemma is equivalent to several other properties that have simpler proofs, see On a lemma of Jacques-Louis Lions and its relation to other fundamental results:

Some of these equivalent properties can be given an independent, i.e., “direct”, proof, such as for instance the constructive proof by M.E. Bogovskii of the surjectivity of the divergence operator. Therefore, the proof of any one of such properties provides, by way of the equivalence theorem, a means of proving J.L. Lions lemma, the known “direct” proofs of which for a general domain are notoriously difficult.

Neither Bogovskii's proof nor the equivalence proof makes use of Fourier analysis, so in that sense this might be what the OP is searching for.

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  • $\begingroup$ It seems the boundedness of the domain important for the equivalence. For if $\Omega = \mathbb R^n$ the space $L^2_0(\Omega)$ would be undefined and the range of $\operatorname{div}: H_0^1(\Omega)\to L^2(\Omega)$ would not be closed. I wonder are there proofs of J.C. Lions lemma that makes no use of Fourier analysis when $\Omega$ unbounded. $\endgroup$
    – Zhang Yuhan
    Commented Mar 25 at 7:32

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