# Does $W^{1,1}([0,1])$ embed isometrically into $L^1([0,1])$

I know there are some results concerning Sobolev spaces compactly embedding into Lebesgue spaces. I'd like to know if $$W^{1,1}([0,1])$$ embeds isometrically into $$L^1$$, or any other Lebesgue space.

Yes, you have the embedding $$T \colon W^{1,1}([0,1]) \to L^1(0,1)^2$$, $$Tf := (f, f').$$
Rellich-Kondrachov theorem is precisely this statement, that $$W^{1,p}(\Omega)\Subset L^{q}(\Omega)$$ for all $$1 \leq q < np/(n-p)$$ where $$\Omega\subset \Bbb R^n$$ is open, bounded and Lipschitz and $$p\in[1,n)$$.