I'd like to clarify the definitions of the spaces involved in the theory of traces of Sobolev functions.

Actually I'm referring to the books of Kufner and Necas.

In these books the boundary integral is introduced using the local description of the boundary of the (Lipschitz) domain $\Omega \subset \mathbb R^N$.

Question 1: I think that one can introduce the space $L^p(\partial \Omega)$ just using the (N-1)-dimensional Hausdorff measure and that, for $0<s<1$, one has

$$ u \in W^{s,p}(\partial \Omega) \iff u \in L^p(\partial \Omega),\; \int_{\partial \Omega} \int_{\partial \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N-1+sp}}\mathrm d \mathcal H^{N-1}(x) \mathrm d \mathcal H^{N-1}(y)<+\infty$$

Is this true?

These are the spaces needed to deal with thaces of first order sobolev functions. For higher order traces one has to introduce the spaces $W^{s,p}(\partial \Omega)$ with $s=m+\sigma$ where $m$ is integer and $o < \sigma<1$. These are defined asking that the compositio of the function $u \in L^p(\partial \Omega)$ with the maps describing the boundary is a Sobolev function on a $N-1$-dimensional domain.

Question 2 Is it really necessary to ask more reguarity to the boundary in order to introduce these spaces? (Kufner defines $W^{s,p}(\partial \Omega)$ only for domains of class $C^{[s],1}$)

Question 3 Any reference where these spaces are treated in detail? I'd like to see that they're well-defined and if they can be endowed with a norm that doesn't depend on the loca description of the boundary.


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