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I recently started the study of Sobolev spaces for my PDE class. In short, I still have quite some difficulties to handle the associated concepts as well as notation. In order to clear things up, I will go along and pose several related questions to the title question.

To start with the initial question, consider the $n$-dimensional unit ball around zero, i.e. $U = B(0,1)$ and the function $$f: U \to \mathbb{R},\; x \mapsto \log(x^2)$$ where $n \geq 2$. I now have to decide for which $p \in [0, \infty]$ $f$ lies in $W^{1, p}(U)$. We defined the space $W^{k, p}$ as $$\{f \in L^1_{loc}(U)\, \vert \, \partial^{\alpha} f \;\text{exists and lies in}\; L^p(U)\; \forall \lvert \alpha\rvert \leq k\}$$

In my case I then have to determine the $p$ for which $f \in L^1_{loc}(U)$ and for which the first partial derivative of $f$ lies in $L^p(U)$, right? Before I present my thoughts so far, one question related to the definition of $W^{k, p}$. What does $\lvert \alpha \rvert$ practically mean? I am aware that it is multiindex notation with $$\lvert \alpha \rvert = \alpha_1 + \ldots + \alpha_n$$ and I also found something here but it seems I cannot really apply it yet. Suppose that we have $$\frac{\partial^2}{\partial x_1^2}, \frac{\partial^3}{\partial x_2^3}, \frac{\partial}{\partial x_3}$$ then we would have $\alpha_1 = 2, \alpha_2 = 3, \alpha_3 = 1$ and hence $\lvert \alpha \rvert = 6$, so we would need to look at $W^{6, p}$. I wonder whether this is right since we only look at the third partial derivative? Obviously, here lies some confusion for me.

Continuing with the exercise, $f$ is smooth when not near $0$ so, at least later, I would have to carve out some $\epsilon$-ball around zero and then somehow estimate it. Furthermore, I would think that $f \in L^1_{loc}(U)$ since, except for $0$ which has measure zero, $\int \lvert f\rvert < \infty$ over $U$. Also, I have $$\partial_{x_i} f = \frac{1}{x^2}{2x_i}$$ Finally, how to proceed next? I guess that I would somehow need to look at $$\int\limits_{U - B(0, \epsilon)} f\phi_{x_i} dx$$ where $\phi \in C_c^{\infty}(U)$ but how do I take the $\epsilon$-ball around zero into account and on then determine the $p \in [0, \infty]$?

Taufi
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  • For the multiindex thing, the point is that $W^{k,p}$ lets you differentiate along any multiindex of size $k$ (or smaller). When you stitch them together in Leibniz notation that means that you'll have $\partial^k$ (or a smaller exponent) in the numerator. – Ian Jun 19 '17 at 15:59
  • I have difficulties grasping what "multiindex of size $k$" means. Why not take $\max{a_i} \leq k$ then? Then you also would have partial derivatives of size $k$ or smaller. – Taufi Jun 19 '17 at 16:50
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    It's the total number of derivatives being at most $k$, not the maximum in each component. – Ian Jun 19 '17 at 16:52
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    So the multiindices of order $3$ consist of those which have either a single $3$ and the rest zeros, or a single $2$, a single $1$, and the rest zeros, or three $1$s and the rest zeros. – Ian Jun 19 '17 at 17:02
  • Okay, now I understand. And if the multiindex is of order 1 then we can only have one, i.e. the first partial derivatives. If it is order 2 then we can have one partial derivative of second order and all other zero or two of order one. Correct? – Taufi Jun 19 '17 at 17:04
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    That's right. ${}$ – Ian Jun 19 '17 at 17:05

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