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

TL;DR

This paper develops a distributional spectral theory for weighted non-local operators in aging media, introducing new weighted function spaces, spectral characterizations, and embeddings that unify fractional regimes within an operator-theoretic framework.

Contribution

It establishes a rigorous distributional framework for weighted non-local operators using transmutation methods, extending Fourier analysis and Sobolev spaces to aging media contexts.

Findings

Constructed the Weighted Schwartz Space and its dual for aging operators.

Extended the Weighted Fourier Transform as a unitary isomorphism.

Derived a sharp embedding theorem linking spectral energy to pointwise 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 $\mathcal{S}_{\psi,\omega}$ 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.