TITLE: Morse lemma with least amount of regularity.
QUESTION [7 upvotes]: I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across with work only for $C^3$ Morse functions. 
A Google search was inconclusive about the existence of a Morse lemma for Morse functions $f: M \to \Bbb R$ with lesser regularity then $C^3$, where $M$ is a smooth finite dimensional manifold.
A reference is perhaps the best possible answer, but any chunk of information will be appreciated.

REPLY [4 votes]: Requiring $C^2$ is too much: you can ask only $C^1$, twice differentiable at the distinguished point with a non-degenerate Hessian matrix. More precisely the following holds true.
Theorem.
Let $\Omega$ be an open subset of $\mathbb R^n$, $x_{0}\in \Omega$ and $f:\Omega\longrightarrow \mathbb R$ be a $C^1$  function twice differentiable at $x_{0}$
such that
$$
df(x_{0})=0,\quad\det f''(x_{0})\not=0.
$$
Then there exist a neighborhood $V$ of $0$ in $\mathbb R^n$,
 a neighborhood $U$ of $x_{0}$ in $\Omega$, and a $C^1$ diffeomorphism $\kappa:V\longrightarrow U$
 such that $\kappa(0)=x_{0},\kappa'(0)=Id,$
 \begin{equation}
(f\circ \kappa)(y)=f(x_{0})+\frac12\langle{ f''(x_{0})y},{y}\rangle.
\end{equation}
For a proof, try your hand or check
Santiago Lopez de Medrano, MR 1378414.