I am working on an example in Evans' PDE book, Sobolev Spaces, Section 5.2.2 Definition of Sobolev Space, Example 3. There is a small part that bothers me. Here is the example and my confusion.
Example $3$. Take $U = B^0(0,1)$, the open unit ball in $\mathbb{R}^n$ and $$ u(x) = |x|^{-\alpha} \quad (x \in U, \ x \neq > 0) $$ For which values of $\alpha > 0,n,p$ does $u$ belong to $W^{1,p}(U)$? (Here $W^{k,p}(U)$ is the Sobolev space on $U$.)
The answer is $$ u \in W^{1,p}(U) \iff \alpha < \frac{n-p}{p} $$ The steps to show this is
When $0\leq\alpha + 1 < n$ we can show (by taking an $\epsilon$ ball and taking the limit) $$ \int_U u\phi_{x_i}dx = - \int_U u_{x_i} \phi dx $$ for all $\phi \in C_c^\infty(U)$
$|Du(x)| = \frac{\alpha}{|x|^{\alpha+1}} \in L^p(U)$ if and only if $(\alpha +1)p < n$
Therefore $u \in W^{1,p}(U)$ if and only if $\alpha < \frac{n-p}{p}$
My question
I am good with step 1. For step 2 and 3, I can show when $\alpha < \frac{n-p}{p}$ the result holds.
What I don't understand is: for the other direction shouldn't I check when does $|D_{x_i}(u)| \in L^p(U)$ hold for $i=1,\cdots,n$? When I look at the definition of Sobolev spaces it says:
"for every multiindex $\beta$ with $|\beta| \leq k$ (in our case $k=1$), $D^\beta u$ exists and belongs to $L^p(U)$"
Instead of this the book checks when does $|Du| \in L^p(U)$ i.e. checks if the gradient of $u$ is $L^p$. Is this somehow equivalent to what I want to check?
Any help is appreciated. Thank you in advance.
Observe \begin{align} \int |\partial_{x_i} u|^p \leq \int |D u|^p \end{align} which mean if the gradient is in $L^p$ then the partials are also in $L^p$.