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Let

  • $d\in\mathbb N$
  • $\Omega\subseteq\mathbb R^d$ be open
  • $\mathcal D(\Omega):=C_c^\infty(\Omega)$
  • $q\ge 1$

I've seen the following Lemma (without a proof) in a paper and don't understand how I need to interpret it:

Let $p\in\mathcal D'(\Omega)$ with $\nabla p\in L^q(\Omega)^d$ $\Rightarrow$ $p\in L_{\text{loc}}^q(\Omega)$.

By definition, $$\nabla p(\Phi)\stackrel{\text{def}}=\sum_{i=1}^d\frac{\partial p}{\partial x_i}(\Phi_i)\stackrel{\text{def}}=-\sum_{i=1}^dp\left(\frac{\partial\Phi_i}{\partial x_i}\right)\;\;\;\text{for all }\Phi\in\mathcal D(\Omega)^d\;.\tag 1$$


I know that each $f\in\mathcal L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal D'(\Omega)$, $$\langle f\rangle(\phi):=\langle\phi,f\rangle_{L^2(\Omega)}\;\;\;\text{for }\phi\in\mathcal D(\Omega)\;.\tag 2$$ I understand that this identification is the meaning of $L^1_{\text{loc}}(\Omega)\subseteq\mathcal D'(\Omega)$. By $(1)$ and $(2)$, we see that $$\nabla\langle f\rangle(\Phi)=-\langle\nabla\cdot\Phi,f\rangle_{L^2(\Omega)}\;\;\;\text{for all }\Phi\in\mathcal D(\Omega)^d\tag 3\;.$$


However, even with $(3)$, I'm not able to make sense of $\nabla p\in L^q(\Omega)^d$. So, what is meant?

[As a secondary question, where can I find a proof of the Lemma and does the Lemma even hold for $q=\infty$?]

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  • $\begingroup$ A distribution $d$ being in $L^p$ means that there exists a function $f$ in $L^p$ such that for $g \in \mathcal{D}$, $d(g)=\int_{-\infty}^\infty f g dx$. $\endgroup$ – Ian Jun 27 '16 at 17:12
  • $\begingroup$ The use of $p$ for both a function and for an index on an $L^p$ space is not helping anything... $\endgroup$ – paul garrett Jun 27 '16 at 17:13
  • $\begingroup$ @paulgarrett Sorry, I've replaced one $p$ by $q$. But Ian made the same mistake by using $d$ for both the space dimension and a distribution ;) $\endgroup$ – 0xbadf00d Jun 27 '16 at 17:16
  • $\begingroup$ Strictly speaking what I said uses a space dimension of $1$ but the extension is clear. $\endgroup$ – Ian Jun 27 '16 at 17:18
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    $\begingroup$ The same is true when you drop the "local" part. It is also true for vector-valued functions: a vector-valued function is in $L^p$ if its norm is in $L^p$. Equivalently, a vector-valued function is in $L^p$ if all its components are in $L^p$. In other words, the meaning of $\nabla p \in L^q(\Omega)^d$ has nothing to do with the fact that $\nabla p$ is the gradient of something. It would have the same meaning if $\nabla p$ were just some arbitrary vector-valued function. $\endgroup$ – Ian Jun 27 '16 at 17:27
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This is standard notation: if $p \in \mathcal D ' (\Omega)$, then $\nabla p \in \mathcal D ' (\Omega, \Bbb R^d)$ is a vector-valued distribution (the terminology is misleading, because it suggests that a vector-valued distribution has vectors as values, which it does not; the term is quite natural, though, in that a vector-valued distribution is a distribution on a space of vector-valued test functions) given by $\langle \nabla p, \varPhi \rangle = - \langle p, \text{div } \varPhi \rangle$.

The definition is quite natural if you think about it for a moment: if $p$ is in fact a smooth function and $F$ a smooth vector-valued map with compact support (i.e. a vector-valued test function), then $\nabla p$ can be viewed as a vector of distributions, and its value on $F$ is (using $\partial _i$ for $\frac {\partial} {\partial x_i}$)

$$\langle \nabla p, F \rangle = \sum _i \langle (\partial_i p), F_i \rangle = \sum _i \int (\partial_i p) F_i = \sum _i \int \partial_i (p F_i) - \sum _i \int p \partial_i F_i = \\ \sum_i 0 - \int p \sum _i \partial_i F_i = - \int p \ \text{div} F = - \langle p, \text{div } F \rangle ,$$

where $\int \partial_i (p F_i) = 0$ because $pF_i$ has compact support in the variable $x_i$ (because $F$ has).

On the other hand, the space $L^q (\Omega)^d$ is the space of vector-valued Lebesgue $q$-integrable functions, i.e. $L^q (\Omega)^d = L^q (\Omega, \Bbb R^d) = \{ V : \Omega \to \Bbb R^d \text{ measurable } \mid \int _\Omega \| V \|^q < \infty \}$, with equality understood as equality almost everywhere (as usual with Lebesgue spaces).

With all this clarified, the notation $\nabla p \in L^q (\Omega) ^d$ means that $\nabla p$ is a vector-valued distribution on $\Bbb R^d$ such that there exist $V \in L^q (\Omega)^d$ with $\nabla p = V$. The equality makes sense because, as with usual distributions, the space $L^q (\Omega) ^d$ embeds naturally in $\mathcal D ' (\Omega, \Bbb R^d)$.

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