Let $S$ the following single layer potential associated to the laplacian problem :
$$S[\varphi](x):= \int_{\partial \Omega} G(x-y) \varphi(y) d\sigma(y) , \quad \varphi \in L^2(\partial \Omega), \quad x \in \partial \Omega$$ where $\Omega$ is a bounded open domain with boundary of class $C^{1,s}$ ($0 < s < 1$) and $G(x):= - \frac{1}{4 \pi |x|}$ is the Green function of the Laplacian. It is well-known that $S$ is a bounded linear operator from $L^2(\partial \Omega)$ to $H^1(\partial \Omega)$, furthermore invertible. Is there a known value or bound on the norm $||S||_{\mathcal{L}(L^2(\partial \Omega), H^1(\partial \Omega))} $ ?
I'm also wondering the same for the double layer potential $D$ defined as
$$D[\varphi](x):= \int_{\partial \Omega} \nabla_y G(x-y) \cdot n(y) \varphi(y) d\sigma(y) , \quad \varphi \in L^2(\partial \Omega), \quad x \in \partial \Omega$$ which is a bounded linear operator from $L^2(\partial \Omega)$ to $L^2(\partial \Omega)$.
I'll be happy to provide further explications if necessary.