Doubt about Sobolev space definition in Evans' book

Question

In the book "Partial Differential Equations, L. Evans". The definition of Sobolev spaces specifies that a function $f \in W^{k,p}$ has to be locally summable (integrable). But, I see multiple times the next definition: \begin{equation} W^{k,p}=\{f \in L^{p}(\Omega) | D^{\alpha}f \in L^{p}(\Omega), \forall \alpha \in \mathbb{N}^{n} \mid |\alpha| \leq k\} \end{equation} Therefore, when the author says "locally summable" it means p-locally summable, $L_{p, loc}(\Omega)$, instead of $L_{1, loc}(\Omega)$? Otherwise, could anyone explain to me what is the relation between both definitions? Because in the first case I can see the relation.

Answer 1

Evans is saying that the space $W^{k,p}(U)$ consists of all locally summable functions $u:U \to \mathbb{R}$, i.e. $u \in L^1_{loc}(U)$, such that $D^\alpha u$, $|\alpha| \leq k$, exists in weak sense and is in $L^p$.

Or in other words:

$$W^{k,p}(U) = \{ u \in L^1_{loc}(U) : D^\alpha u \in L^p(U), |\alpha| \leq k \}.$$

This set is the same as $X=\{ u \in L^p(U) : D^\alpha u \in L^p(U), |\alpha|\leq k \}.$ The direction $X \subset W^{k,p}(U)$ is clear since $L^p(U) \subset L_{loc}^1(U)$. The other direction follows if you choose the index $\alpha=(0,\dots,0)$, which gives $u=D^\alpha u \in L^p$.