On a class of globally hypoelliptic operators

We consider an operator $A$ which is defined on an $(n+1)$-dimensional manifold $\Omega$ and which is elliptic everywhere outside an $n$-dimensional submanifold $\Gamma$. If $(x)$ represents the local coordinates in $\Gamma$ and $t$ is the distance to $\Gamma$, then in the coordinates $(x,t)$ the operator $A$ is of the form
$$ Au=\sum_{|\beta|+l\leqslant m}a_{\beta l}(x,t)t^{lq}D^\beta_xD^l_tu, $$
where $q>1$ is an integer. We present a necessary and sufficient condition for infinite differentiability in a neighborhood of $\Gamma$ of the solution of $Au=f$ if $f$ is infinitely differentiable in a neighborhood of $\Gamma$.
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