In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet $p$-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

中文翻译：

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一般向量场的$ p $-次Laplacians主频

在本文中，我们证明了一般向量场的Dirichlet $ p $ -sub-Laplacian主频率（或第一个特征值）的唯一性和简单性。作为副产品，我们建立了Caccioppoli不等式，并在Grushin平面和Heisenberg组上讨论了特殊情况。