I need a statement that must be classical but it's being very difficult (even after consulting experts) to find a reference. Let's try here!

Let $f$ be a function on the boundary of a bounded domain $\Omega$ in $\mathbb R^n$ and let $w$ be the single layer potential. Namely, $w(x)=\int_{\partial\Omega} f(y) P(x-y) dH^{n-1}$, where $P$ is the Newtonian potential.

If the domain is $C^{k, \alpha}$ then it should be true...

  • $f\in C^{k-1,\alpha}$ $\Rightarrow$ $w\in C^{k,\alpha}(\overline{\Omega})\cap C^{k,\alpha}(\overline{\Omega^c}) $ ($k+\alpha$ derivatives up to the boundary for each side)

  • $f\in C^{k-2,\alpha}$ $\Rightarrow$ $\frac 1 2 (\partial_{\nu, {\rm in}}+ \partial_{\nu, {\rm out}}) w \in C^{k-1,\alpha}(\partial\Omega)$, where $\partial_{\nu, {\rm in}}$, $\partial_{\nu, {\rm out}}$ denote the normal derivatives form inside an outside. (we should obtain an extra gain do to the typical cancelation...)

This type of result in for $k=1$ is contained in classical books of potential theory and PDE from math physics (Sobolev, Miranda, ...). I would like to find (if it exists) a clean modern reference for all $k$. It is possible that one needs to flatten de boundary and then apply a general enough result (which I do not know) on boundedness of Riesz-type transforms...


The first result is stated on p.303 of

Dautray, R., Lions, J-L, Mathematical analysis and numerical methods for science 
and technology. Vol. 1. Physical origins and classical methods. 
Springer-Verlag, Berlin, 1990.

The full proof is not given but the authors say :

''By combining the geometrical techniques used in the proofs of Propositions 10, 11 and 12 and the singular integrals techniques used in the proof of Proposition 9 we can demonstrate that...''


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