weak derivatives of exp(-|x|) and Hilbert Spaces

Question

To which Hilbert Space (W^m,2) of R does the function exp(-|x|) belongs? I know its weak derivative is (-exp(-x) for x>0, exp(x) for x<0 and c0 (arbitrary) for x = 0). This weak derivative is in L2, but does it have a weak derivative also?

Note: the doamin is the real line R

Answer 1

The function $f(x)=e^{-|x|}$  has a weak derivative $f'(x)=-{\rm sgn\,}(x)\,e^{-|x|}$, while $f,f'\in L^2(\mathbb{R})$. Hence $f\in W^{1,2}(\mathbb{R})$. But $f'$ does not have a weak derivative, since its disribution derivative $f''=e^{-|x|}+2\delta\, e^{-|x|}=e^{-|x|}+2\delta$  in $\mathcal{D}'(\mathbb{R})$. Hence $f\notin W^{m,2}(\mathbb{R})\;\forall\,m\geqslant 2$.

Answer 2

The function $f(x)=\mathrm{e}^{-x}$ does not belong to any $W^{m,2}(\mathbb R)$, $m\ge 0$, because if it did, then we would have that $f\in L^2(\mathbb R)$, which is not true since $$ \int_{-\infty}^\infty |f(x)|^2\,dx=\int_{-\infty}^\infty \mathrm{e}^{-2x}\,dx=\infty. $$