# Weak derivative and Locally summable functions

I have three question regarding the appearance of the space of locally summable functions in the definition of weak derivatives and sobolev spaces.

The deifinition of weak derivatives from Evans:

Suppose $$v, u \in L^1_{loc}(U)$$, and $$\alpha$$ is a multiindex. We say that v is the $$\alpha$$:th weak partial derivative of $$u$$, written $$D^{\alpha}u=v$$, provided

$$\int_U uD^\alpha\phi dx =(-1)^{|\alpha|}\int_U v\phi dx$$, for all test functions $$\phi \in C^\infty_c(U)$$

1) Why is it necessary to use the space $$L^1_{loc}(U)$$ and not for example $$L^1(U)$$?

In the definition of a Sobolev space $$W^{k,p}(U)$$ in Evans:

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

As I understand, locally summable means that $$u \in L^1_{loc}(U)$$.

2) Does the usage of $$L^1_{loc}(U)$$ in this definition comes from the fact that it is used in the definition of the weak derivative above?

Last, the Sobolev norm $$\|u\|_{W^{k,p}(U)}$$ is defined as:

$$\|u\|_{W^{k,p}(U)} = (\sum_{ |\alpha| \leq k}\int_U|D^\alpha u|^p dx)^{\frac{1}{p}}$$,

3) But for a function $$u \in W^{k,p}(U)$$ how do we now that the term $$\int_U|u|^p dx$$ (that appears in the Sobolev norm when $$\alpha = (0,0, ...,0)$$) is $$<\infty$$ since all we know is that $$u \in L^1_{loc}(U)$$ and we dont have that $$u \in L^p (U)$$?

Thanks :)

I had a similar question while reading through Evans. I don't know the answer to all your questions, but I think I can respond to some of them.

"As I understand, locally summable means that $$u \in L^1_{loc}(U)$$."

Yes, this is correct. One can verify by looking at the Wikipedia page for locally summable functions, for example.

As for your question 2, I think this is true. It's a bit strange to me because most books I have seen do not have the locally summable part, but maybe it's a bit more general.

As for your question 3, the full definition of a Sobolev space in the book is that the Sobolev space $$W^{k, p}(U)$$ consists of all locally summable functions $$u: U \to \mathbb{R}$$ s.t. for each multiindex $$\alpha$$ with $$|\alpha| \leq k$$, $$D^\alpha u$$ exists in the weak sense and belongs to $$L^p(U)$$.

So taking $$\alpha = 0$$ in the definition is exactly what you want.

The answer to 1) is that in order to make sense of the expression $$\int u \phi$$ for $$\phi$$ a $$C^\infty$$ function with compact support, it is enough that $$u \in L_\mathrm{loc}^1$$ (you do not need that $$u \in L^1$$).

What is the added value of allowing $$u$$ to be only in $$L_\mathrm{loc}^1$$ instead of asking that $$u \in L^1$$? Well, you want to be able to say that weak derivatives are a generalization of usual derivatives. But $$C^1$$ functions are not in general in $$L^1(\mathbb R^n)$$, though they are always in $$L_\mathrm{loc}^1(\mathbb R^n)$$. So if you want to talk about the weak derivative $$u(x)=x^2$$ on $$\mathbb R$$, and prove that it is equal to $$v(x)=2x$$, you need to work with $$L_\mathrm{loc}^1(\mathbb R)$$.