Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods

The spectral analysis of operators in heterogeneous and aging media typically requires a functional framework that extends beyond the standard Hilbertian setting. In this paper, we establish a rigorous distributional theory for a class of non-local operators, termed Weighted Weyl-Sonine operators, by employing a structure-preserving transmutation method. We construct the Weighted Schwartz Space ${\mathcal{S}}_{\psi ,\omega }$ and its topological dual, the space of Weighted Tempered Distributions ${\mathcal{S}}_{\psi ,\omega }^{\mathrm{\prime }}$, ensuring that the underlying Fréchet topology is consistent with the infinitesimal generator of the aging dynamics. This topological foundation allows us to: (i) extend the Weighted Fourier Transform to generalized functions as a unitary isomorphism; (ii) provide an explicit spectral characterization of the weighted Dirac delta ${\delta }_{\psi ,\omega }$ and its scaling laws under geometric dilations; and (iii) introduce a scale of Weighted Sobolev Spaces $H_{\psi ,\omega }^s$ defined via spectral multipliers. A central result is the derivation of a sharp embedding theorem, $\mathrm{\mid }u(t)\mathrm{\mid }\le C\omega (t{)}^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{H_{\psi ,\omega }^s}$, which rigorously connects abstract spectral energy to the pointwise decay induced by the weight $\omega$. This framework provides a unified geometric characterization of several fractional regimes, including the Hadamard and Riemann-Liouville cases, within a single operator-theoretic architecture.