AI 中文总结
研究原点处有幂型奇点的光滑函数及全纯函数的分数阶1/2拉普拉斯算子，基于特定测试函数空间定义，通过对洛朗级数应用构造得含雅可比多项式的新级数，还推导了相关应用及泛函演算。
AI 中文摘要
我们给出了原点处具有幂型奇点的光滑函数的1/2分数阶拉普拉斯算子的分布定义，包括负单项式和某些具有孤立奇点的全纯函数。该构造基于与测试函数的利佐金空间相关的合适测试函数空间，其测试函数及其分数阶拉普拉斯算子都属于施瓦茨类且在原点处无穷阶消失，从而补偿原点处的任何幂型奇点。这使得1/2分数阶拉普拉斯算子可通过对偶性定义，而无需基础函数满足通常的可积性假设。所考虑的分布框架在1/2分数阶拉普拉斯算子下是不变的，产生了自然的半群性质。通过对洛朗级数逐项应用该构造，我们得到了涉及雅可比多项式的新级数。通过此过程，我们为一类全纯函数定义了分数阶拉普拉斯算子并推导了几个应用，特别强调了最近发展的关于S谱上谱理论的四元数精细结构理论，描述了在由富埃特 - 谢 - 钱延拓定理产生的各类全纯型函数中扩展经典全纯函数演算的泛函演算。
英文摘要
We give a distributional definition of the $\frac{1}{2}$-fractional Laplacian for smooth functions with power-type singularities at the origin, including negative monomials and certain holomorphic functions with isolated singularities. The construction is based on a suitable space of test functions related to the Lizorkin space of test functions, for which both the test functions and their fractional Laplacians belong to the Schwarz class and vanish to infinite order at the origin, and thus compensate for any singularity of power type at the origin. This allows the $\frac{1}{2}$-fractional Laplacian to be defined by duality without requiring the underlying function to satisfy the usual integrability assumptions. The considered distributional framework is shown to be invariant under the $\frac{1}{2}$-fractional Laplacian, yielding a natural semigroup property. By applying the construction term by term to Laurent series we obtain new series that, remarkably, involve Jacobi polynomials. With this procedure we define the fractional Laplacian for a class of holomorphic functions and derive several applications, with particular emphasis on the recently developed theory of the quaternionic fine structures of the spectral theory on the $S$-spectrum, describing the functional calculi extending the classical holomorphic functional calculus in the various classes of holomorphic-type functions arising from the Fueter-Sce-Qian extension theorem.
