A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators

Abstract: Standard fractional calculus has successfully modeled systems with power-law memory. However, complex phenomena in heterogeneous media often exhibit multi-scale memory effects and aging properties that classical operators cannot capture. In this work, we construct a unified framework by defining the \textit{Weighted Weyl-Sonine Operators}. This formalism offers a fundamental generalization of fractional calculus, freeing the theory from the constraints of power-law memory (via Sonine kernels), time-translation invariance (via scale and weight functions), and artificial history truncation (via Weyl integration). The main result is a Generalized Spectral Mapping Theorem, proving that the Weighted Fourier Transform acts as a universal diagonalization map for these operators. We rigorously characterize the admissible memory kernels through the class of \textit{Complete Bernstein Functions}, ensuring that the resulting operators preserve the fundamental properties of positivity and monotonicity. Furthermore, we establish a theoretical bridge between the algebraic Sonine definition and the analytical Marchaud representation involving Lévy measures. Finally, we apply this theory to solve generalized relaxation equations and \textit{Weighted Distributed Order} evolution problems, demonstrating that phenomena of ultra-slow diffusion and retarded aging can be treated explicitly within this unified spectral framework.