Analytic hypoellipticity of Keldysh operators

We consider Keldysh-type operators, P = x1D2 x1 + a(x)Dx1 + Q(x, Dx ), x = (x1, x ) with analytic coefficients, and with Q(x, Dx ) second order, principally real and elliptic in Dx for x near zero. We show that if P u = f, u ∈ C∞, 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 in (Lebeau and Zworski, Proc. Amer. Math. Soc. 147 (2019) 145–152; Zworski, Bull. Math. Sci. 7 (2017) 1–85) and has applications to scattering theory