About the definition of Sobolev Spaces

Question

I'm studying Sobolev Space and I have a question about the definition:

Def.: The Sobolev Space $W^{k,p}(U)$ consists of all locally summable functions $u:U\to \mathbb{R}$ such that for each multiindex $\alpha$ with $|\alpha|\geq k,$ $D^\alpha u$ exist in the weak sense and belongs to $L^p(U).$

Observation: If $k=0$ and $p=2$, then $W^{0,2}(U)=L^2(U)$.

We have that $ u\in W^{k,p}(U)$ since $u\in L^1_{loc}(U),$ so every $u\in L^2(U)$ belong to $L^1_{loc}(U)?$

How can I show that? Thanks.

Answer 1

Indeed this is just Holder's inequality: Pick $V \subset U$ with $|V|<\infty$, then

$$||u||_{L^1(V)}=\int_V |u| dx\le \sqrt{\int_V 1^2 dx}\sqrt{\int_V |u|^2 dx} = \sqrt{|V|}\ ||u||_{L^2(U)}$$

then $u \in L^1_{loc}(U)$.