I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$.
$\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set of all functions $f\in L_2(0,1)$ for which
$|| f(\cdot + h) - f(\cdot)||_{L_2(0,1-h)} \le C h^{\alpha}$ for every $1 > h > 0$.
I thought about $f(x) = x^{\alpha}$ which is in $L(\alpha, L_{\infty}(0,1))$ but not in higher spaces. But some numerical experiments show that this function is possibly in $\mathrm{Lip}(1,L_2)$ and not only in $\mathrm{Lip}(\alpha, L_2(0,1))$.
Can someone help me?