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## Trace theorem for $C^{k,1}$ domains

What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains?

For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known that the trace map is bounded (and has a continuous right inverse) from $H^{s}(\Omega)$ to $H^{s-1/2}(\partial\Omega)$ for $s\in(\frac12,\frac32)$. Moreover the endpoint case $s=\frac32$ is claimed by David Jerison and Carlos Kenig but a proof seems to have not appeared. A part of the question is whether or not the claimed proof appeared. The other part is if there is a proof of the obvious extension of this result to $C^{k,1}$ domains. If so, what is the situation of the corresponding endpoint (i.e., $s=k+\frac32$) result?

Update: As pointed out in the answers, there is a paper by Doyoon Kim which completely answers the second part of my question. An earlier paper by Jürgen Marschall also answers this question, with possible exception of some exponents (interestingly, this paper appeared before Costabel's paper). I say "possible", because the both papers treat the traces for $L^p$-based spaces and Marschall's paper exclude the case $s-\frac1p$ is an integer for the boundedness of the right inverse. But his methods maybe applicable for $p=2$ (I have not checked) as we know this case is somewhat special. In any case Kim proves the theorem even when $s-\frac1p$ is an integer, for general $p$.

Now the remaining question is the endpoint case $s=k+\frac32$.

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I also believe the Kim paper cited above has the relevant result. On a related matter, there's a very nice paper by Buffa, Costabel and Sheen in J.Math.Anal.Appl. (2002) on trace theorems for H(curl) fields in Lipschitz domains. And Luc Tartar may have the relevant result you seek for part 2 of your question.

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The abstract of the following paper sounds like it might be relevant to your first question. Unfortunately, I do not have access to the journal: D. Kim, Trace theorems for Sobolev-Slobedeckij spaces with or without weights, Journal of Function Spaces and Applications 5 (2007), 243-268.

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The aforementioned article is now available from here.

Abstract:

We prove that the well-known trace theorem for weighted Sobolev spaces holds true under minimal regularity assumptions on the domain. Using this result, we prove the existence of a bounded linear right inverse of the trace operator for Sobolev-Slobodeckij spaces $W_p^s(\Omega)$ when $s-1/p$ is an integer.
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