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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).

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

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..)

Let $\Omega$ be a bounded Lipschitz domain.

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

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

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).

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

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..)

Let $\Omega$ be a bounded Lipschitz domain.

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

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