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18.104
\\
Spring 2013
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\begin{center}\textbf{Homework 6}\end{center}

\begin{center}Due March 5th by the beginning of class.\end{center}

\textbf{Problem:} Prove that $u \in \mathscr D'(\mathbb R^2)$ solves the system
\begin{equation}\label{e:system}
(x_1^2 - x_2^2)u = 0, \qquad x_1x_2 u = 0,
\end{equation}
if and only if there exist $c_0,c_1,c_2, c_3 \in \mathbb C$ such that
\[
u = c_0 \delta + c_1 \partial_1 \delta + c_2 \partial_2 \delta + c_3(\partial_1^2+\partial_2^2)\delta.
\]
\textit{Hint:} Deduce $x_1^3 u =0$ and $x_2^3 u =0$ from \eqref{e:system}, and show these imply $u = \sum_{|\alpha|\le N} c_\alpha \partial^\alpha \delta$ for some $N \in \mathbb N$. The additional restrictions on the $c_\alpha$ follow from these two equations, as well as the individual equations in \eqref{e:system} (here it may be helpful to use the formulas for the coefficients $c_\alpha$ provided at the end of the proof of Theorem 3.2.1 in \cite{frjo}).

\textbf{Solution:}


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

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