  A classical result of Malgrange says that for a polynomial P and an open
subset $\Omega$ of $\R^d$ the differential operator $P(D)$ is surjective on
$C^\infty(\Omega)$ if and only if $\Omega$ is P-convex. H\"ormander showed that
$P(D)$ is surjective as an operator on $\mathscr{D}'(\Omega)$ if and only if
$\Omega$ is strongly P-convex. It is well known that the natural question
whether these two notions coincide has to be answered in the negative in
general. However, Tr\`eves conjectured that in the case of d=2 P-convexity and
strong P-convexity are equivalent. A proof of this conjecture is given in this
note.
