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][1] and [Zhonghai Ding][2], 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][3] 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][4] by [Doyoon Kim][5] which completely answers the second part of my question. An earlier [paper][6] 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 $p$.

**Update 2:** 
The first part of the question, that is the claim about the endpoint case $s=k+\frac32$ seems to be not true. A counterexample due to Guy David has appeared in a [1995 paper by Jerison and Kenig][7] on the inhomogeneous Dirichlet problem.


  [1]: http://www.ams.org/mathscinet-getitem?mr=937473
  [2]: http://www.ams.org/mathscinet-getitem?mr=1301021
  [3]: http://www.ams.org/mathscinet-getitem?mr=716504
  [4]: http://www.ams.org/mathscinet-getitem?mr=2352844
  [5]: http://dkim.khu.ac.kr/
  [6]: http://www.ams.org/mathscinet-getitem?mr=884984
  [7]: http://www.ams.org/mathscinet-getitem?mr=1331981