Return to Question

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 $p$.

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 on the inhomogeneous Dirichlet problem.

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: Essentially complete answer has been found, and recorded below.

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 $s=k+\frac32$.

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 $p$.

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 on the inhomogeneous Dirichlet problem.

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, 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 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?

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 $s=k+\frac32$.