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18.104
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Spring 2013
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\begin{center}\textbf{Homework 8}\end{center}

\begin{center}Due March 12th by the beginning of class.\end{center}

\textbf{Problem:} Prove that $u \in \mathscr D'(\mathbb R^n)$ solves $x_nu =0$ if and only if $u=v(x')\otimes \delta(x_n)$ for some $v \in \mathscr D'(\mathbb R^{n-1})$. (Notation is as in \cite[Theorem 4.3.4]{frjo}.)

\textit{Hint:} Fix $\chi \in C_c^\infty(\mathbb R)$ with $\chi(0)=1$. Show that for any $\phi \in C_c^\infty(\mathbb R^n)$ there is $\tilde \phi \in C_c^\infty(\mathbb R^n)$ such that $\phi(x) = \phi(x',0)\chi(x_n) + x_n \tilde\phi(x)$ (this is similar to the $m=1$ case of \cite[(2.7.7)]{frjo}). Proceed as in the proof of \cite[Theorem 4.3.4]{frjo}.

\textbf{Solution:}


\begin{thebibliography}{00}
\bibitem[FrJo]{frjo} G. Friedlander and M. Joshi. The Theory of Distributions, second edition, Cambridge University Press, 1998.
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