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

Question

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.

Answer 1

Yes, you have the embedding $T \colon W^{1,1}([0,1]) \to L^1(0,1)^2$, $$ Tf := (f, f'). $$

Answer 2

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 $p\in[1,n)$.