AI 中文总结
本文构造分数阶算子代数，通过有限规范形和因果反演，将平坦卷积算子与多项式乘子统一，并处理弱奇异方程及Prabhakar核的范围刻画。
AI 中文摘要
分数阶算子代数为将代数规范形与因果反演联系起来提供了自然框架，但任意的光滑乘子和Volterra核会破坏有限分数阶重写。我们构造了一个由多项式乘子、以包含$\mathbb{Z}$的加法子群$\Gamma\subset\mathbb{R}$为指标的分数幂，以及平坦因果卷积算子生成的代数。其有限多项式加权对角平坦卷积算子构成包含平坦生成元的最小双边理想，多项式乘子是内蕴的，分布分离产生具有加法阶过滤的唯一有限规范形。对于首一规范形，首项幂选择了一个典范因果右因子，该因子从平坦核心延伸到自然范围$J^\alpha C[0,T]$。在$L^1$中完成负阶卷积扇区可接纳可积的Sonine和Prabhakar核，而独立给定的连续二元Volterra项保持外部。在单射性和幂型核界条件下，可变低阶项和该二元扰动可化为一个弱奇异第二类方程，其经典预解式在每个有限区间上存在，无需小性假设。一个有限边界射流块纳入给定的初始数据而不改变约化核。Sonine、分布阶和正则化Prabhakar实现随之而来。在非整数Prabhakar机制中，一个精确的范围准则刻画了经典$AC^m$子域，一个显式的Riemann-Liouville例子证明它严格小于连续自然范围。
英文摘要
Fractional operator algebras provide a natural setting for linking algebraic normal forms to causal inversion, but arbitrary smooth multipliers and Volterra kernels destroy finite fractional rewriting. We construct an algebra generated by polynomial multipliers, fractional powers indexed by an additive subgroup $Γ\subset\mathbb{R}$ containing $\mathbb{Z}$, and flat causal convolution operators. Its finite polynomially weighted diagonal-flat convolution operators form the smallest two-sided ideal containing the flat generators, polynomial multipliers are intrinsic, and distributional separation yields a unique finite normal form with an additive order filtration. For monic normal forms, the leading power selects a canonical causal right factor that extends from the flat core to the natural range $J^αC[0,T]$. Completing the negative-order convolution sector in $L^1$ admits integrable Sonine and Prabhakar kernels, while an independently prescribed continuous bivariate Volterra term remains external. Under injectivity and power-type kernel bounds, variable lower-order terms and this bivariate perturbation reduce to one weakly singular second-kind equation, whose classical resolvent exists on every finite interval without smallness assumptions. A finite boundary-jet block incorporates prescribed initial data without changing the reduced kernel. Sonine, distributed-order, and regularized-Prabhakar realizations follow. In the noninteger Prabhakar regime, an exact range criterion characterizes the classical $AC^m$ subdomain, and an explicit Riemann--Liouville example proves that it is strictly smaller than the continuous natural range.
Comments32 pages
