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.