Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917231381708800 |
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| author | Dorrego, Gustavo |
| author_facet | Dorrego, Gustavo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21497 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods Dorrego, Gustavo Functional Analysis Analysis of PDEs Classical Analysis and ODEs 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. |
| title | Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2601.21497 |