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.



Morse lemma with least amount of regularity.
You only need $C^2$. See Nirenberg's book *Topics in Nonlinear Functional Analysis*, Theorem 3.1.1. He attributes this version of the Morse lemma to the late great Lars Hormander, *Fourier Integral Operators I*.


