高阶Grushin算子的Carleman估计
Carleman estimates for higher step Grushin operators
AI总结：
本文针对在子流形上退化的高阶Grushin算子，利用Gegenbauer多项式加法公式推导加权$L^p-L^q$估计以建立Carleman型不等式，并由此证明了Schrödinger算子在退化点处的强唯一延拓性质。
摘要：
高阶Grushin算子$Δ_α$是一族在$\mathbb{R}^{n+m}$的子流形上退化的次椭圆算子。本文为这些算子建立了Carleman型不等式。这是通过推导Grushin调和投影子的加权$L^{p}-L^{q}$估计实现的。证明中的关键要素是T. Koornwinder和Y. Xu给出的Gegenbauer多项式加法公式。作为推论，我们得到了Schrödinger算子$-Δ_α+V$在退化流形点处的强唯一延拓性质，其中$V$属于某个$L^{r}_{\rm loc}(\mathbb{R}^{n+m})$。
英文摘要：
The higher step Grushin operators $Δ_α$ are a family of sub-elliptic operators which degenerate on a sub-manifold of $\mathbb{R}^{n+m}$. This paper establishes Carleman-type inequalities for these operators. It is achieved by deriving a weighted $L^{p}-L^{q}$ estimate for the Grushin-harmonic projector. The crucial ingredient in the proof is the addition formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a consequence, we obtain the strong unique continuation property for the Schrödinger operators $-Δ_α+V$ at points of the degeneracy manifold, where $V$ belongs to certain $ L^{r}_{\rm loc}(\mathbb{R}^{n+m})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
