Schwartz theorem

Can i relax the conditions on f of the Schwartz theorem in real analysis?

That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the same.Can i ask for something less than $C^2(\Omega)$? Like $f \in H^2(\Omega)$, that is the space of functions in $L^2(\Omega)$ which also have first and second derivative in $L^2(\Omega)$.