I am reading Evans' book on Partial differential equations. Where he proves the existence of trace operator $T:W^{1,p}(\Omega) \rightarrow L^p(\Omega),$ for $1 \leq p <\infty.$
Does there exist such an operator for fractional Sobolev spaces or Sobolev spaces with negative order?
If yes, where can I find the details?
If not, why is it not possible to define the traces?
Best Answer
A simple case to understand why it is sometimes not possible to define traces is when the space is $L^p$. Then the functions are defined only almost everywhere so the trace has no meaning. On the other hand, a meaning can easily be given when a the space embeds in the space of continuous functions (which is the case of $W^{s,p}(\mathbb R^d)$ when $s > d/p$).
For the case $p=2$, in the book of L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, you will find a chapter about trace for $H^s$ spaces when $s>1/2$. In the case $s=1/2$, there is an interesting special space $H^{1/2}_{0,0}$.
More generally, you can find the result in Triebel's book Theory of Function Spaces II, Section 4.4. If you do not know Besov and Triebel Lizorkin generalizations of Sobolev's spaces, you just have to know that
Then the result is that if $p\in[1,\infty]$ and $s> 1/p$, then the trace is continuous from $F^s_{p,q}$ to $F^{s-1/p}_{p,p}$.