About fractional Sobolev space

Question

I'm reading a paper used Sobolev space $H^s(\Omega)$ , I only know the definition of these space when $\Omega=R^n$ which used Fourier Transform, what if $\Omega$ is a bounded open set?

And I also want to know the bounded linear functional on this space look like?

Any suggestions are welcome!

Answer 1

Maybe you want to take a look to this paper: http://arxiv.org/abs/1104.4345