Analytic hypoellipticity of Keldysh operators

For operators modeled by P = x1D2 x1 +D2 x2 +aDx1 we show that if u is smooth and Pu is analytic then u is analytic. This is motivated by the question of analyticity of quasinormal modes of black holes across event horizons and is a consequence of a general microlocal result. Joint work with J Galkowski.