Let $\Omega$ be a bounded Lipschitz domain.

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

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).

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