Heisenberg椭圆算子和横截Heisenberg椭圆算子的指标理论：$KK$-理论视角
Index theory for Heisenberg elliptic and transversally Heisenberg elliptic operators from $KK$-theoretic viewpoint
AI总结：
本研究从$KK$-理论视角，运用Kasparov方法系统研究了Heisenberg椭圆及横截Heisenberg椭圆算子的指标理论，为亚椭圆微分算子分析提供了稳健框架。
摘要：
本研究全面描述了横截Heisenberg椭圆算子的基本理论，并从$KK$-理论的视角，运用Kasparov的方法，研究了Heisenberg椭圆算子和横截Heisenberg椭圆算子的指标理论。此外，分析方法系统地考察了特定条件，重点关注幂零群$C^{\ast}$-代数的傅里叶变换。我们展示了分析算子亚椭圆性的增强方法，在$KK$-理论框架下为定义和理解横截Heisenberg椭圆性提供了稳健的框架。这项工作为未来研究复杂流形中亚椭圆微分算子的性质奠定了坚实基础。
英文摘要：
This research comprehensively describes the basic theory of transversally Heisenberg elliptic operators, and investigates the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators from the perspective of $KK$-theory, applying Kasparov's methodology. Moreover, the analysis methodically examines specific conditions, with a focus on the Fourier transform of the nilpotent group $C^{\ast}$-algebra. We demonstrate enhanced methods for analyzing the hypoellipticity of operators, presenting a robust framework for defining and understanding transversal Heisenberg ellipticity in a $KK$-theoretic context. This work provides a solid foundation for future research into the properties of hypoelliptic differential operators in complicated manifolds.
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