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It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.

For simplicity take $\Omega = (0,\infty)\times\mathbb{R}$ the half plane, so $\partial\Omega=\{0\}\times\mathbb{R}$. Is it also true that the trace operator is surjective from $T:W^{2,1}(\Omega)\to W^{1,1}(\partial\Omega)$ and if not what is its image?

The proof of surjectivity for $T:W^{1,1}(\Omega) \to L^1(\partial\Omega)$ does not extend to $W^{2,1}(\Omega)$, as the second derivative in the normal direction to $\partial\Omega$ fails to be in $L^1(\Omega)$. In fact, my intuition is that this condition implies that the image of $T:W^{2,1}(\Omega)\to W^{1,1}(\partial\Omega)$ is a weakly compact subset of $W^{1,1}(\partial\Omega)$. Is this true?

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    $\begingroup$ I believe that it's not surjection. For p>1 and s>1/p the image given by Besov space, which collapse to L^p at the endpoint. I believe the similar reverse result via Besov spaces should holds for p=1. Do you have a quick reference for the first paragraph. $\endgroup$
    – Liding Yao
    Commented Apr 14 at 17:53
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    $\begingroup$ I also believe it is not surjective. I could try to write a counterexample, but I am overwhelmed now with the end of the semester. $\endgroup$
    – Piotr Hajlasz
    Commented Apr 14 at 23:03
  • $\begingroup$ The surjectivity in the W^{1,1} case is a theorem by Gagliardo, and was asked previously mathoverflow.net/questions/145393/… which includes references. Having worked on it more I think the image of the trace on $W^{s,1}$ is the Besov space $B^{s-1}_{1,1}$ for all $s>1$ as you suggested, though a full proof still eludes me. $\endgroup$
    – vmist
    Commented Apr 14 at 23:40
  • $\begingroup$ @LidingYao do you mean the $s=\frac{1}{p}$ endpoint? i.e. that the trace is surjective $T: W^{\frac{1}{p},p}(\Omega) \to L^p(\partial\Omega)$ for $p>1$? And if so I would love a reference. $\endgroup$
    – vmist
    Commented Apr 14 at 23:57
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    $\begingroup$ @vmist The first google result gives me "Trace Operators in Besov and Triebel-Lizorkin Spaces" by Cornelia Schneider in 2010. It mentions the $p>1$ in the beginning. My guess of $\operatorname{Tr}W^{2,1}$ is $B_{11}^1$. Note that this doesn't imply by any Besov-Triebel-Lizorkin cases since $W^{2,1}\not\subset F_{1\infty}2$ so it needs a careful proof. Certainly Prof. Hajlasz is more expert on this than me. $\endgroup$
    – Liding Yao
    Commented Apr 15 at 1:28

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