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 $\mathcal{S}'_{\psi,\omega}$, ensuring that the underlying Fr\'echet 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 $H^{s}_{\psi,\omega}$ defined via spectral multipliers. A central result is the derivation of a sharp embedding theorem, $|u(t)| \le C \omega(t)^{-1} \|u\|_{H^s_{\psi,\omega}}$, 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.