one can define the Schauder spaces also on $M$ by exploiting the local parametrizations. In particular, one can consider the spaces $C^{k,\alpha}(\partial\Omega)$ on $\partial\Omega$ for $0\leq k\leq m$ with $\Omega$ a bounded open set of class $C^{m,\alpha}$, and the trace operator from $C^{k,\alpha}({\mathrm{cl}}\Omega)$ to $C^{k,\alpha}(\partial\Omega)$ is linear and continuous. We denote by $d\sigma$ the area element of a manifold imbedded in ${\mathbb{R}}^{n}$. Also, we find convenient to set $$C^{k,\alpha}(\partial \Omega)_{0}\equiv
\left\{
f\in C^{k,\alpha}(\partial \Omega):\,\int_{\partial\Omega}f\,d\sigma=0 
\right\}\,.$$ For the definition and properties of real analytic maps, we refer to Deimling [@De85 p. 150]. In particular, we mention that the pointwise product in Schauder spaces is bilinear and continuous, and thus real analytic (cf., *e.g.*, Lanza de Cristoforis and Rossi [@LaRo04 pp. 141, 142]).

Let $S_{n}$ be the function from ${\mathbb{R}}^{n}\setminus\{0\}$ to ${\mathbb{R}}$ defined by $$S_{n}(x)\equiv
\left\{
\begin{array}{lll}
\frac{1}{s_{n}}\log |x| \qquad &   \forall x\in 
{\mathbb{R}}^{n}\setminus\{0\},\quad & {\mathrm{if}}\ n=2\,,
\\
\frac{1}{(2-n)s_{n}}|x|^{2-n}\qquad &   \forall x\in 
{\mathbb{R}}^{n}\setminus\{0\},\quad & {\mathrm{if}}\ n>2\,,
\end{array}
\right.$$ where $s_{n}$ denotes the $(n-1)$-dimensional measure of $\partial{\mathbb{B}}_{n}(0,1)$. $S_{n}$ is well-known to be a fundamental solution of the Laplace operator.

We now introduce the simple layer potential. Let $\alpha\in]0,1[$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$. If $\mu\in C^{0}(\partial\Omega)$, 