This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including$L_a=\text{div}({\left|y\right|}^a\mathrm{\nabla }),$ with $a\in (-1,1)$ and their perturbations.

As they belong to the Muckenhoupt class $A_2$, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni [1], [2], [3] and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension [4]. Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin [5], [6], [7].

这项工作致力于对节点退化集或奇异方程组的解的几何理论分析，涉及一类算子，包括${\mathrm{大号}}_{\mathrm{一种}}=\text{div}（{\left|ÿ\right|}^{\mathrm{一种}}\mathrm{\nabla }），$ 与 $\mathrm{一种}\in （-\mathrm{1个}，\mathrm{1个}）$ 和他们的摄动。

因为它们属于Muckenhoupt类 ${\mathrm{一种}}_2$，这些运算符出现在Fabes，Kenig，Jerison和Serapioni [1]，[2]，[3]的开创性著作中，由于它们与分数拉普拉斯算子的本地化联系，最近在最近十年中引起了很多关注。通过扩展扩展[4]。我们的目标是根据Hardt，Simon，Han和Lin [5]，[6]，[7]的开创性工作，为此类方程的节点解集的分层性质建立完整的理论。 ]。