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 $\mathcal{S}{ψ,ω}$ 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.

💡 Research Summary

The paper develops a comprehensive functional‑analytic framework for a broad class of non‑local operators called Weighted Weyl‑Sonine operators, which arise in heterogeneous and aging media. The authors begin by fixing a smooth, strictly increasing diffeomorphism ψ : ℝ→ℝ and a positive smooth weight ω : ℝ→(0,∞) with at most polynomial growth. Using these, they define a unitary transmutation operator
 T : L²_{ψ,ω}(ℝ) → L²(ℝ), (Tf)(y)=ω(ψ⁻¹(y)) f(ψ⁻¹(y)).
T simultaneously accounts for the geometric stretch ψ and the density weight ω. Because T is an isometry, the classical Schwartz space 𝒮(ℝ) can be pulled back to a weighted Schwartz space
 𝒮_{ψ,ω}=T⁻¹𝒮(ℝ).
The topology of 𝒮_{ψ,ω} is generated by seminorms ρ_{k,m}(φ)=sup_{t∈ℝ}|ψ(t)^{k} D_{ψ,ω}^{m}φ(t)|, where the weighted derivative
 D_{ψ,ω}φ(t)=\frac{1}{ω(t)ψ’(t)}\frac{d}{dt}\bigl(ω(t)φ(t)\bigr)
plays the role of the infinitesimal generator of the Weyl‑Sonine family. The space is Fréchet, dense in the weighted Hilbert space L²_{ψ,ω}, and admits an orthonormal basis of weighted Hermite functions H_{ψ,ω}^{n}(t)=ω(t)^{-1}h_{n}(ψ(t)).

The dual space 𝒮’{ψ,ω} is defined via the pairing
 ⟨T,φ⟩{ψ,ω}=⟨T^{-1}T, T^{-1}φ⟩{L²},
ensuring compatibility with the transmutation map. Within this distributional setting the weighted Dirac delta is identified explicitly as
 δ{ψ,ω}(t−τ)=\frac{1}{ω(τ)^{2}ψ’(τ)} δ(t−τ).
The factor (ω²ψ’)⁻¹ reflects how the medium’s density and time‑scale dilation dilute or amplify an impulsive source.

The weighted Fourier transform is introduced as the composition F_{ψ,ω}=F∘T, where F is the classical Fourier transform. Because T is unitary, F_{ψ,ω} is a unitary isomorphism on 𝒮_{ψ,ω} and extends to 𝒮’{ψ,ω} by duality: for any distribution T, its transform bT satisfies ⟨bT,ϕ⟩{ψ,ω}=⟨T,F_{ψ,ω}ϕ⟩{ψ,ω}. This definition preserves Parseval’s identity and allows the spectral diagonalization of weighted convolution kernels. In particular, a Weyl‑Sonine operator with Bernstein symbol Φ(iξ) satisfies the spectral mapping
 D{ψ,ω} ↔ Φ(iξ)·F_{ψ,ω}u,
so that the operator becomes multiplication by Φ(iξ) in the transformed domain.

Sobolev regularity is introduced via the spectral norm
 ‖u‖{H^{s}{ψ,ω}}^{2}=∫{ℝ}(1+|ξ|^{2})^{s}|F{ψ,ω}u(ξ)|^{2}dξ.
The transmutation operator restricts to a unitary map T:H^{s}{ψ,ω}→H^{s}(ℝ), so all classical Sobolev results carry over. The central embedding theorem states that for s>½ there is a continuous embedding
 H^{s}{ψ,ω} ↪ C^{0}{ψ,ω}(ℝ)
with the explicit pointwise bound
 |u(t)| ≤ C{s} ω(t)^{-1}‖u‖{H^{s}{ψ,ω}} for all t∈ℝ.
Thus the weight ω acts as a physical envelope: larger ω forces stronger decay of the signal amplitude.

The authors apply this machinery to fractional diffusion equations of the form
 (D^{α}{ψ,ω}+I)u=f, α∈(0,2).
If f∈H^{s}{ψ,ω}, then the unique solution satisfies u∈H^{s+α}{ψ,ω} and the operator (D^{α}{ψ,ω}+I) is a topological isomorphism between these spaces. The proof relies on the transformed algebraic equation ((iξ)^{α}+1)·b u(ξ)=b f(ξ) and the fact that the symbol never vanishes, guaranteeing boundedness of the inverse multiplier.

For impulsive forcing, the paper studies the Green’s function G_{t₀} solving (D^{α}{ψ,ω}+I)G{t₀}=δ_{ψ,ω}(·−t₀) with α>1. Using the Sobolev embedding, G_{t₀} belongs to H^{s+α}{ψ,ω} for s<−½ and satisfies the decay estimate
 |G{t₀}(t)| ≤ C ω(t) ω(t₀).
This double weighting illustrates how both the source time and observation time are moderated by the medium’s density.

Finally, the framework subsumes classical fractional calculi. Setting ψ(t)=ln t (t>0) and ω≡1 recovers the Hadamard fractional derivative, for which the embedding reduces to a uniform bound (since ω⁻¹=1). The authors emphasize that the pair (ψ,ω) serves as a dictionary translating regularity results across different fractional regimes, including Riemann–Liouville and other multi‑scale models.

In summary, by employing a “structure transport” via the transmutation operator, the paper builds a weighted Schwartz–Sobolev hierarchy that is fully compatible with the spectral theory of Weighted Weyl‑Sonine operators. This unified approach resolves the limitations of the Hilbert‑space-only setting, accommodates singular inputs, and provides sharp pointwise decay estimates dictated by the weight ω, thereby offering a powerful tool for analysis of aging, heterogeneous, and fractional diffusion phenomena.