$C^{m}(\Omega )$ of those functions $f$ whose derivatives $D^{\eta }f$ of order $|\eta |\leq m$ can be extended with continuity to $\mathrm{cl}\Omega$ is denoted $C^{m}(
\mathrm{cl}\Omega )$. The subspace of $C^{m}(\mathrm{cl}\Omega ) $ whose functions have $m$-th order derivatives that are uniformly Hölder continuous with exponent $\alpha\in
]0,1]$ is denoted $C^{m,\alpha} (\mathrm{cl}\Omega )$ (cf., *e.g.*, Gilbarg and Trudinger [@GiTr83]). The subspace of $C^{m}(\mathrm{cl}\Omega ) $ of those functions $f$ such that $f_{|{\mathrm{cl}}(\Omega\cap{\mathbb{B}}_{n}(0,R))}\in
C^{m,\alpha}({\mathrm{cl}}(\Omega\cap{\mathbb{B}}_{n}(0,R)))$ for all $R\in]0,+\infty[$ is denoted $C^{m,\alpha}_{{\mathrm{loc}}}(\mathrm{cl}\Omega ) $.

Now let $\Omega $ be a bounded open subset of ${\mathbb{R}}^{n}$. Then $C^{m}(\mathrm{cl}\Omega )$ and $C^{m,\alpha }({\mathrm{cl}}
\Omega )$ are endowed with their usual norm and are well-known to be Banach spaces. We say that a bounded open subset $\Omega$ of ${\mathbb{R}}^{n}$ is of class $C^{m}$ or of class $C^{m,\alpha}$, if ${\mathrm{cl}}
\Omega$ is a manifold with boundary imbedded in ${\mathbb{R}}^{n}$ of class $C^{m}$ or $C^{m,\alpha}$, respectively (cf., *e.g.*, Gilbarg and Trudinger [@GiTr83 §6.2]). We denote by $
\nu_{\Omega}
$ the outward unit normal to $\partial\Omega$. For standard properties of functions in Schauder spaces, we refer the reader to Gilbarg and Trudinger [@GiTr83].

If $M$ is a manifold imbedded in ${\mathbb{R}}^{n}$ of class $C^{m,\alpha}$, with $m\geq 1$, $\alpha\in ]0,1[$, 