Analytic hypoellipticity of Keldysh operators

We consider Keldysh-type operators, $P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x'} )$, $x = ( x_1, x')$ with analytic coefficients, and with $Q ( x, D_{x'} )$ second order, principally real and elliptic in $D_{x'}$ for $x$ near zero. We show that if $P u =f$, $u \in C^\infty$, and $f$ is analytic in a neighbourhood of $0$ then $u$ is analytic in a neighbourhood of $0$. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and has applications to scattering theory.