I'm kind of stuck in a Problem and hope that one of you is able to resolve it.
Setting of the Problem:
Let $\Omega \subseteq \mathbb{R}^3$ be a bounded $\mathscr{C}^2$-domain (this is just a technical assumption to ease the problem I'm dealing with) and let $H^1(\mathbb{R}^3,\mathbb{C}^4)$ be the space of functions who's components are Sobolev-functions. Then the free Dirac operator is defined by:
$\text{dom}(A_0) := H^1(\mathbb{R}^3,\mathbb{C}^4)$
$A_0 f := (-ic \alpha*\nabla +mc^2\beta )f \hspace{0,3cm}$ for all $\hspace{0,3cm}f \in \text{dom}(A_0)$
(Here $\alpha$ is the vector consiting of the Dirac matrices and $\beta$ is a matrix aswell) It's well known that $A_0$ is self adjoint in $L^2(\mathbb{R}^3,\mathbb{C}^4)$ and it's spectrum is given by $\sigma(A_0) = (-\infty,-mc^2] \cup [mc^2,\infty)$. Moreover there's an explicit integralrepresentation of the resolvent for $\lambda \in \rho(A_0)$:
$(A_0-\lambda)^{-1}f(x) = \int_\limits{\mathbb{R}^3} G_{\lambda}(x-y)f(y) dy\hspace{0,3cm}$ for all $\hspace{0,3cm}f \in L^2(\mathbb{R}^3,\mathbb{C}^4)$ and $x \in \Omega$
with the integral kernel
$G_{\lambda}(x) = \left( \frac{\lambda}{c^2} I_4+m \beta \left( 1- i \sqrt{\frac{\lambda^2}{c^2}-m^2c^2} ||x||\right) \frac{i \alpha*x}{c ||x||^2}\right)\frac{exp \left(i \sqrt{\frac{\lambda^2}{c^2}-m^2c^2} ||x||\right)}{4 \pi ||x||}$
(Here the square root is choosen such that $Im(\sqrt{.}) > 0$ for all $\lambda \in \rho(A_0)$)
Now since $(A_0-\lambda)^{-1}f \in H^1(\mathbb{R}^3,\mathbb{C}^4)$ for all $f \in L^2(\mathbb{R}^3,\mathbb{C}^4)$ we have for the restriction $(A_0-\lambda)^{-1}f \big|_{\Omega} \in H^1(\Omega,\mathbb{C}^4)$.
Since $\Omega \subseteq \mathbb{R}^3$ is a bounded $\mathscr{C}^2$-domain, there exists the trace operator $\gamma : H^1(\Omega,\mathbb{C}^4) \rightarrow H^{\frac{1}{2}}(\partial \Omega,\mathbb{C}^4)$ and we have $\gamma \left( (A_0-\lambda)^{-1}f \big|_{\Omega} \right) \in H^{\frac{1}{2}}(\partial \Omega,\mathbb{C}^4)$
The actuall problem: Starting from the line above I have to derive an explicit integral representation of $\gamma \left( (A_0-\lambda)^{-1}f \big|_{\Omega} \right)$. I found a paper which treats a simmilar problem and the author just states that $\gamma \left( (A_0-\lambda)^{-1}f \big|_{\Omega} \right)(x) = \int_\limits{\mathbb{R}^3} G_{\lambda}(x-y)f(y) dy\hspace{0,3cm}$ for all $\hspace{0,3cm}x \in \partial \Omega$. They just insert $x \in \partial \Omega$ into the resolvent formula of the free Dirac operator to define a function in $L^2(\partial \Omega, \mathbb{C}^4)$.
My question now: is it really that simple and am I missing something? When I was studing BEM we actually proved that for the Newtonian potential
$(N_0 f)(x) := \int_\limits{\mathbb{R}^3} \frac{f(y)}{4 \pi ||x-y||} dy \hspace{0,3cm}$ for all $\hspace{0,3cm} f \in L^2(\mathbb{R}^3,\mathbb{C})$ and $x \in \mathbb{R}^3$
we have that it's trace is given by: $\gamma \left( N_0 f \right) (x) = \int_\limits{\mathbb{R}^3} \frac{f(y)}{4 \pi ||x-y||} dy \hspace{0,3cm}$ for all $\hspace{0,3cm} f \in L^{\infty}(\mathbb{R}^3,\mathbb{C})$ and $x \in \partial \Omega$. So the integral representation is only valid for $L^{\infty}$-functions and I can't see why the integral representation of the trace should be vaild for all $L^{2}(\mathbb{R}^3,\mathbb{C}^4)$-functions.
My attempt so far: I was thinking of showing a integral representaion of $\gamma \left( (A_0-\lambda)^{-1}f \big|_{\Omega} \right)$ for $\mathscr{C}^{\infty}_{c}(\mathbb{R}^3,\mathbb{C}^4)$-functions, use the fact that all $L^2(\mathbb{R}^3,\mathbb{C}^4)$-functions can be approximated by $\mathscr{C}^{\infty}_{c}(\mathbb{R}^3,\mathbb{C}^4)$-functions in the $L^2$-norm and use that the Operator $ f \mapsto \gamma \left( (A_0-\lambda)^{-1}f \big|_{\Omega} \right)$ is a bounded and linear operator from $L^2(\mathbb{R}^3,\mathbb{C}^4)$ to $L^2(\partial \Omega,\mathbb{C}^4)$. So to speak copy the proof of the one for the trace of the Newtonian potential, but it seems to be kind of messy doing this ...
I hope someone has an alternativ approach or a hint how to deal with these kind of problems. Moreover if you have a book or other reference for problems involving potentials and traces I would be grateful.
Thanks in advance, GordonFreeman