Mathematical Research Letters

Volume 10 (2003)

Number 6

On a question of Louis Nirenberg

Pages: 729 – 735



François Treves (Rutgers University)


This note proves that if $A,B$\ are $\mathcal{C}^{\infty }$\ real vector fields in an open set $\Omega \subset \mathbb{R}^{3}$\textbf{\ }such that $% A,B$\ \ and $[A,B]$\ are linearly independent then,\ given any $\mathcal{C}% ^{\infty }$\ real vector field $C$\ in $\Omega $\ and any function $\varphi \in $\textbf{\ }$\mathcal{C}^{\infty }\left( \Omega \right) $, the second order operator $L=AB+C+\varphi $\ is locally solvable at every point of% \textbf{\ }$\Omega $\textbf{.} The result can be extended to first-order real pseudodifferential operators with simple real characteristics.

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