We consider Keldysh-type operators, $P=x_1{D}_{x_1}^2+a(x)D_{x_1}+Q(x,D_{x^{\prime }})$, $x=(x_1,x^{\prime })$ with analytic coefficients, and with $Q(x,D_{x^{\prime }})$ second order, principally real and elliptic in $D_{x^{\prime }}$ for $x$ near zero. We show that if $Pu=f$, $u\in C^{\infty }$, 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 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.

中文翻译：

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Keldysh 算子的解析次椭圆度

我们考虑 Keldysh 类型的运算符， $磷=X_1{D}_{X_1}^2+\mathrm{一种}(X)D_{X_1}+问(X,D_{X^{\prime }})$, $X=(X_1,X^{\prime })$ 与解析系数，并与 $问(X,D_{X^{\prime }})$ 二阶，主要是实数和椭圆 $D_{X^{\prime }}$ 为了 $X$接近于零。我们证明如果$磷你=F$, $你\in C^{\infty }$， 和 $F$是分析在附近$0$， 然后 $你$在 0 的邻域中是解析的。这是微局部结果的结果，对于具有拉格朗日径向集的任何阶的算子都有效。我们的结果证明了 (Lebeau and Zworski, Proc. Amer. Math. Soc . 147 (2019) 145–152; Zworski, Bull. Math. Sci . 7 (2017) 1–85)中的猜想的广义版本，并且具有散射理论的应用