We prove a multiplier theorem of Mihlin-Hormander type for operators of the form $-\Delta_x - V(x) \Delta_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, where $V(x) = \sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2\sigma}$, and $\sigma \in (1/2,\infty)$. The result is sharp whenever $d_1 \geq \sigma d_2$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrodinger operators, which are stable under perturbations of the potential.

中文翻译：

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用于 Grushin 算子的锐乘子定理的稳健方法

我们在 $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2} 上证明了形式为 $-\Delta_x - V(x) \Delta_y$ 的运算符的 Mihlin-Hormander 类型的乘数定理_y$，其中$V(x) = \sum_{j=1}^{d_1} V_j(x_j)$，$V_j$ 是幂律的扰动$t \mapsto |t|^{2\sigma} $, 和 $\sigma \in (1/2,\infty)$。每当 $d_1 \geq \sigma d_2$ 时，结果就会很明显。结果的主要新颖之处在于其稳健性：这似乎是非椭圆亚椭圆算子的第一个尖锐乘法器定理，允许步长大于 2 和系数扰动。证明取决于对一维薛定谔算子的特征值和特征函数的精确估计，它们在势能的扰动下是稳定的。