In 1971 Fediĭ proved in [3] the remarkable theorem that the linear second order partial differential operator$L_{f}u(x,y)\equiv \{\frac{\partial }{\partial x^2}+f{\left(x\right)}^2\frac{\partial }{\partial y^2}\}u(x,y)$is hypoelliptic provided that $f\in C^{\infty }\left(\mathbb{R}\right)$, $f\left(0\right)=0$ and f is positive on $(-\infty ,0)\cup (0,\infty )$. Variants of this result, with hypoellipticity replaced by continuity of weak solutions, were recently given by the authors, together with Cristian Rios and Ruipeng Shen, in [16] to infinitely degenerate elliptic divergence form equations${\mathrm{\nabla }}^{\mathrm{tr}}\mathcal{A}(x,u)\mathrm{\nabla }u=\varphi \left(x\right),x\in \mathrm{\Omega }\subset {\mathbb{R}}^n,$where the nonnegative matrix $\mathcal{A}(x,u)$ has bounded measurable coefficients with trace roughly 1 and determinant comparable to $f^2$, and where $F=\ln \frac{1}{f}$ is essentially doubling.

However, in the plane, these variants assumed additional geometric constraints on f, such as $f\left(r\right)\ge e^{-r^{-\sigma }}$ for some $0<\sigma <1$, something not required in Fediĭ's theorem. In this paper we in particular remove these additional geometric constraints in the plane for homogeneous equations, and only assume that f is positive away from the origin.

1971年Fediĭ在[3]中证明了线性二阶偏微分算子的显着定理${升}_F你(X,是)\equiv \{\frac{\partial }{\partial X^2}+F{\left(X\right)}^2\frac{\partial }{\partial {是}^2}\}你(X,是)$是亚椭圆的，条件是$F\in C^{\infty }\left(\mathbb{电阻}\right)$, $F\left(0\right)=0$并且f为正$(-\infty ,0)\cup (0,\infty )$. 最近，作者与 Cristian Rios 和 Ruipeng Shen 在 [16] 中给出了这个结果的变体，用弱解的连续性代替了亚椭圆度，以无限退化椭圆发散形成方程${\mathrm{\nabla }}^{\mathrm{tr}}\mathcal{一种}(X,你)\mathrm{\nabla }你=\phi \left(X\right),X\in \mathrm{\Omega }\subset {\mathbb{电阻}}^n,$其中非负矩阵 $\mathcal{一种}(X,你)$具有有界的可测量系数，迹线大致为 1，行列式可与$F^2$，以及哪里 $F=\mathrm{输入}\frac{1}{F}$ 本质上是翻倍。

然而，在平面中，这些变体假设了对f 的额外几何约束，例如$F\left(r\right)\ge {\mathrm{电子}}^{-r^{-\sigma }}$ 对于一些 $0<\sigma <1$，Fediĭ 定理中不需要的东西。在本文中，我们特别去除了齐次方程平面中的这些额外几何约束，并且仅假设f远离原点为正。