I'm looking for the differential dimension of $W^{l,p}(\mathbb{R^n})$, defined as follows:
Given a function space $Z(\mathbb{R^n})$, we say that $\mu\in \mathbb{R}$ is the differential dimension of $Z(\mathbb{R^n})$ if for each $f\in Z(\mathbb{R^n})$ (with compact support), exists $\epsilon_0>0$ and $c_1, c_2>0$ such that
$$c_1\epsilon^{\mu}\|f\|_{Z(\mathbb{R^n})}\leq \|f(\epsilon x)\|_{Z(\mathbb{R^n})}\leq c_2\epsilon^{\mu}\|f\|_{Z(\mathbb{R^n})},$$ for every $\epsilon\geq \epsilon_0$.
I have just proved that the homogeneous Sobolev space $w^{l,p}(\mathbb{R^n})$ and $L^p(\mathbb{R^n})$ have differential dimension $l-n/p$ and $-n/p$, respectively. Using these facts I found that $\|f(\epsilon x)\|_{W^{l,p}(\mathbb{R^n})}\leq \epsilon^{l-n/p}\|f\|_{W^{l,p}(\mathbb{R^n})}$, but I cannot check the second inequality of the definition.
Can anyone help me? Thanks.