Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods
Gustavo Dorrego
TL;DR
This work develops a distributional framework for generalized aging operators in heterogeneous media by transferring the classical Schwartz space structure through a transmutation operator $\mathcal{T}$, yielding weighted spaces $\mathcal{S}_{\psi,\omega}$ and $\mathcal{S}'_{\psi,\omega}$. The Weighted Fourier Transform extends unitarily to distributions, enabling spectral diagonalization of non-local aging operators and a precise description of impulses via $\delta_{\psi,\omega}$. A scale of Weighted Sobolev spaces $H^s_{\psi,\omega}$ is introduced with a sharp pointwise embedding $|u(t)| \le \frac{C_s}{\omega(t)} \|u\|_{H^s_{\psi,\omega}}$, connecting spectral energy to physical decay and encompassing Hadamard and RL fractional regimes within one operator-theoretic framework. This functional-analytic foundation supports well-posedness analysis for aging diffusion problems and paves the way for numerical schemes in singular control contexts by clarifying how medium geometry governs both source normalization and solution decay.
Abstract
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 $\mathcal{S}_{ψ,ω}$ and its topological dual, the space of Weighted Tempered Distributions $\mathcal{S}'_{ψ,ω}$, 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 $δ_{ψ,ω}$ and its scaling laws under geometric dilations; and (iii) introduce a scale of Weighted Sobolev Spaces $H^{s}_{ψ,ω}$ defined via spectral multipliers. A central result is the derivation of a sharp embedding theorem, $|u(t)| \le C ω(t)^{-1} \|u\|_{H^s_{ψ,ω}}$, which rigorously connects abstract spectral energy to the pointwise decay induced by the weight $ω$. This framework provides a unified geometric characterization of several fractional regimes, including the Hadamard and Riemann-Liouville cases, within a single operator-theoretic architecture.