Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.


How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?


I define $H^{\frac 1 2}(\partial\Omega)$ to be space of functions $u \in L^2(\partial\Omega)$ such that
$$|u|\_{H^{\frac 12}(\partial\Omega)} = \int\_{\partial\Omega}\int\_{\partial\Omega} \frac{|u(x)-u(y)|^2}{|x-y|^n} < \infty$$
and give it the norm 
$$\lVert \cdot \rVert\_{H^{\frac 12}(\partial\Omega)}^2 = \lVert \cdot \rVert\_{L^2(\partial\Omega)}^2 + |\cdot|\_{H^{\frac 12}(\partial\Omega)}^2$$


There is the following result:
(1) If $D \subset \mathbb{R}^n$ is an open Lipschitz domain with bounded boundary then $H^1(D) \subset H^{\frac 12}(D)$ is continuous. 


We can use an alternative norm on $H^{\frac 12}(\partial\Omega)$ by using chart maps (which are Lipschitz) to map neighbourhoods of the boundary onto subsets $D\_i$ of Euclidean space. But AFAIK we do not know that these subsets $D\_i$ are Lipschitz, so we cannot apply result (1).


Edit: A reference to this result would be great too.


(I ask here because my question on M.SE did not receive attention, hope that is OK..)



Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Lions and Magenes define $H^\frac12$ by complex interpolation, hence your desired property holds by assumption; but they always assume $\Omega$ to have smooth boundary.


In the books of Adams and Grisvard you will find some related results, but - as far as I have (quickly) seen - not exactly what you look like.


