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This exercise is 6.19 in Rudin's Functional analysis.

The Problem:

$\Lambda \in \mathscr{D}'(\Omega), \ \phi \in \mathscr{D}(\Omega), \ (D^{\alpha}\phi)(x)=0, \ \forall \ x$ in the support of $\Lambda$ and multi-index $\alpha$, show that $\Lambda \phi=0$

Similar to Theorem 6.25 in Rudin (The proposition is that a distribution whose support is one point $p$ is a linear combination of $\delta$ and its derivatives at $p$), I can prove the case that the support of $\Lambda$ is compact, but I don't know how to generalize to the general case.

Thank you very much!

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If you can prove the case that $\Lambda$ is of compact support, it is good. Then for the general case, suppose that $\psi\in C_c^\infty(\Omega)$ such that $\text{supp}(\phi)\subset\text{supp}(\psi)$ and $\psi(x)=1$ on $\text{supp}(\phi)$. You can prove that $\text{supp}(\psi\Lambda)\subset \text{supp}(\psi)$, so $\psi\Lambda\in \mathcal{D}'(\Omega)$ is of compact support. Also, you can show that $\langle \psi\Lambda,D^\alpha\phi\rangle=\langle \Lambda,D^\alpha\phi\rangle$ for all $\alpha$. Then by the first part, $\langle \Lambda,\phi\rangle=\langle \psi\Lambda,\phi\rangle=0$.

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