I'm reading Figalli's proof of Sard's theorem, here is the link of the paper.
In the first step of the proof, at page 3 of the paper, there are two statements for which I cannot find a reference theorem that validates them. Here are the statements:
Since $Y\in W^{k,p}(V)$, $DY$ is invertible, $k\geq 2$ and $p>n\geq k$, it is simple to verify that $X=Y^{-1}$ is still $W^{k,p}$.
and
For $k\geq 2$ and $p>n\geq k$ the composition of functions in $W^{k,p}_{loc}$ is still $W^{k,p}_{loc}$
Now clearly these two statements seems very resonable, but I wonder if there are same classic results that rigorously justify them.
The second statement is Theorem 2 in the paper Bourdaud, G.. Le calcul fonctionnel dans les espaces de Sobolev. Invent. Math. 104 (1991), 435–446. You can also find it in the second edition of Leoni
I don’t have a reference for the first. Using Morrey embedding theorem, you know that the function $Y$ is of class $C^{k-1}$. So you only need to prove that the derivatives of order $k-1$ admit a weak derivative. I think that what Figalli suggests is to take $X(Y(x))=x$, differentiate both sides to get $DX(Y(x))DY(x)=I$ (I am probably missing some transpose), multiply from the left by the inverse of $DY(x)$. This gives you a formula for $DX$. But then you have to differentiate $k-1$ times and maybe use Faa di Bruno formula. It is a bit of a mess. I believe the result is true but definitely not easy.