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In Evan's book, the definition of Sobolev space $W^{k,p}(U)$ consists of all locally integrable function such that for each multiindex $\alpha$ with $0\leq|\alpha|\leq k$, $D^\alpha u$ exists in the weak sense and belongs to $L^p(U)$.

I am wondering why we don't we say it consists of all $L^P$ integrable functons instead of all locally integrable function. We require $D^0u \in L^p$, $u$ must be locally integrable. enter image description here

Velobos
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000000000
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  • I haven't seen it defined that way. I always see it defined as the set of $v\in L^p$ such that $D^\alpha v\in L^p$ for all $0\le |\alpha| \le k$ as you say. –  Sep 10 '20 at 05:52
  • yeah, I checked Brezis' book, the definition is $L^p$ as well. I am not sure if I have missed something. – 000000000 Sep 10 '20 at 06:04
  • https://math.stackexchange.com/questions/3398977/doubt-about-sobolev-space-definition-in-evans-book – 000000000 Sep 12 '20 at 06:38

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