I was recently reading a paper (Lumley, R. R., & Zervos, M. (2001). A Model for Investments in the Natural Resource Industry with Switching Costs.) that contains the following statement:
Elements of the Sobolev space $W^{2,p}_{loc}((0,\infty))$, $p\geq 1$, have sufficient smoothness to be able to apply Ito's formula.
Where can I find a proof of this result? There is no reference in the paper. It was also mentioned that $W^{2, \infty}_{loc}((0,\infty))$ can be identified with the function space $C^{1, 1}_{loc}((0,\infty))$, whose elements are continuously differentiable functions with locally Lipschitz first derivatives. What does it mean that 2 spaces can be identified with each other?