# Proof of the Helmholtz-Hodge decomposition

Let

• $\Omega\subseteq\mathbb R^3$ be open
• $\mathcal D(\Omega):=C_c^\infty(\Omega)$

Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$ and $$H^2(\Omega):=G^2(\Omega)^\perp\;.\tag 0$$ I want to show that $$H^2(\Omega)=\overline{\mathfrak D(\Omega)}^{\left\|\;\cdot\;\right\|_{L^2(\Omega,\;\mathbb R^3)}}\;\;\;\text{with }\mathfrak D(\Omega):=\left\{\phi\in\mathcal D(\Omega)^3:\nabla\cdot\phi=0\right\}\;.\tag 1$$

I found this statement in a seminar paper. I know almost nothing about distribution theory, so please be patient. By definition, $$\nabla f(\phi)\stackrel{\text{def}}=\sum_{i=1}^3\frac{\partial f}{\partial x_i}(\phi_i)\stackrel{\text{def}}=-\sum_{i=1}^3f\left(\frac{\partial\phi_i}{\partial x_i}\right)\;\;\;\text{for all }\phi\in\mathcal D(\Omega)^3\tag 2$$ for all $f\in\mathcal D'(\Omega)$.

I know that each $p\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle p\rangle\in\mathcal D'(\Omega)$, $$\langle p\rangle(\phi):=\langle\phi,p\rangle_{L^2(\Omega)}\;\;\;\text{for }\phi\in\mathcal D(\Omega)\;.\tag 3$$ I understand that this identification is the meaning of $L^1_{\text{loc}}(\Omega)\subseteq\mathcal D'(\Omega)$. By $(2)$ and $(3)$, we see that $$\nabla\langle p\rangle(\phi)=-\langle\nabla\cdot\phi,p\rangle_{L^2(\Omega)}\;\;\;\text{for all }\phi\in\mathcal D(\Omega)^3\tag 4\;.$$

Question:

1. Let $p\in L_{\text{loc}}^2(\Omega)$ and $\phi\in\mathfrak D(\Omega)$. The author of the paper starts the proof (at the beginning of page 2) by showing that $$\nabla\langle p\rangle(\phi)=0\;,\tag 5$$ if $\nabla\langle p\rangle\in G^2(\Omega)$, i.e. if $$\nabla\langle p\rangle=\langle u\rangle$$ for some $u\in L^2(\Omega)^3$. However, we don't need $\nabla\langle p\rangle\in G^2(\Omega)$, since $(5)$ immediately follows from the definition of $\mathfrak D(\Omega)$ and $(4)$. Is there anything I'm missing?
2. What have we actually proved by $(5)$? I think that I ask this question, cause I don't understand what $(0)$ actually means. Since $G^2(\Omega)$ is a space of distributions, I don't understand how its orthogonal complement is defined.
3. Why is the author of the paper considering the special case of space dimension $d=3$ and ($q=2$)-th power integrable functions? Shouldn't at least the given statement hold for any $d\in\mathbb N$ and $q\ge 1$ (or even $q=\infty$)?