Abstract
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.
Highlights
We consider analytic regularity for generalizations of the Keldysh operator [24],P := x1Dx21 + Dx22 . (1.1)The operator P has the feature of changing from an elliptic to a hyperbolic operator at x1 = 0
It appears in various places including the study of transsonic flows, see, for instance, Canic– Keyfitz [8] or population biology — see Epstein–Mazzeo [12]
Our interest in such operators comes from the work of Vasy [31] where the transition at x1 = 0 corresponds to the boundary at infinity for asymptotically hyperbolic manifolds, crossing the event horizons of Schwartzschild black holes or the cosmological horizon for de Sitter spaces
Summary
We consider analytic regularity for generalizations of the Keldysh operator [24],. P := x1Dx21 + Dx22. The operator P has the feature of changing from an elliptic to a hyperbolic operator at x1 = 0 It appears in various places including the study of transsonic flows, see, for instance, Canic– Keyfitz [8] or population biology — see Epstein–Mazzeo [12]. Our interest in such operators comes from the work of Vasy [31] where the transition at x1 = 0 corresponds to the boundary at infinity for asymptotically hyperbolic manifolds (see [34]), crossing the event horizons of Schwartzschild black holes (see [11, § 5.7]) or the cosmological horizon for de Sitter spaces. We prove this result for generalized Keldysh operators with analytic coefficients (1.3). References, and connections to several complex variables, see Christ [9] and for some recent progress and additional references, Bove–Mughetti [7]
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