May 25, 2026
This paper establishes a rigorous functional analytic framework for weighted Weyl-Sonine fractional operators on semi-infinite intervals. While the classical Phillips functional calculus relies strictly on completely monotonic Bernstein functions to subordinate linear semigroups, we demonstrate that this structural restriction can be entirely bypassed. By exploiting the intrinsic topological decay induced by an admissible amnesia weight (\(\omega\)), we prove that the abstract integral representation converges absolutely for general classes of tempered Sonine kernels, thereby unifying the treatment of both diffusive relaxation and oscillatory non-local dynamics within a single framework. Employing the theory of strongly continuous semigroups, we characterize these operators as the infinitesimal generators of a deformed translation semigroup that simultaneously incorporates subjective aging scales (\(\psi\)) and history-attenuating amnesia effects. This similarity-based approach not only provides a sound abstract foundation for the hypersingular Marchaud representation of these operators but also guarantees their closedness and the analytical well-posedness of the associated abstract fractional Cauchy problems in general Banach spaces. Furthermore, we introduce the associated generalized fractional integral and prove the fundamental inversion theorem, demonstrating the complete analytical symmetry of the theory. Finally, we establish pointwise decay estimates for the solutions via weighted Sobolev embeddings.
In this work, we transition from the classical point-wise integral representation to a generator-based semigroup approach. Building upon the operational definition of the Weighted Weyl-Sonine derivative introduced in [1], we now provide its formal functional analytic foundation. Rather than relying on classical subordination extensions—which often impose strict analytic constraints on the memory kernels—we exploit the intrinsic topological decay provided by the weighted functional spaces. We demonstrate that the amnesia weight naturally neutralizes the asymptotic growth of the kernels, thereby establishing a framework that completely bypasses the traditional requirement of complete monotonicity (Bernstein functions). This allows for the mathematically rigorous accommodation of general admissible Sonine kernels, unifying the treatment of both diffusive and oscillatory non-local dynamics.
The study of Sonine kernels has recently experienced a major resurgence due to their central role in general fractional calculus [2]. Recent applications have spanned from scale-invariant systems via Mellin convolutions [3] to advanced Petrov-Galerkin numerical schemes [4]; however, these approaches remain predominantly anchored to finite intervals or specific algebraic settings. Unlike recent studies on generalized Sonine kernels [5]–[7], which are restricted to the Riemann-Liouville or Caputo causal framework on the finite interval \([0, T]\), this work extends the scope of these operators to encompass the full history of the system. By defining the Weighted Weyl-Sonine derivative on the semi-infinite interval \((-\infty, t]\), we capture the long-term hereditary effects characteristic of aging media. The convergence of these non-local integrals at \(-\infty\), a classic hurdle in Weyl-type calculus, is rigorously ensured here by the amnesia weight \(\omega(t)\), which acts as a topological regulator within our transmuted functional framework.
The rest of this paper is organized as follows: Section 2 introduces the preliminary functional spaces, the subjective time weights, and the properties of generalized Sonine kernels. Section 3 constructs the deformed translation semigroup and formally characterizes its infinitesimal generator. In Section 4, we prove the equivalence between the abstract semigroup representation and the hypersingular Marchaud formulation of the Weighted Weyl-Sonine derivative; moreover, we introduce the associated generalized fractional integral and establish the fundamental inversion theorem. Finally, Section 5 establishes the well-posedness of the abstract Cauchy problem via the Hille-Yosida theorem and derives physical pointwise decay estimates using weighted Sobolev embeddings [8].
We briefly recall the standard definitions from the theory of strongly continuous semigroups (\(\mathcal{C}_0\)-semigroups) on a Banach space \((X, \|\cdot\|_X)\). A family of bounded linear operators \(\{T(s)\}_{s \ge 0}\) is a \(\mathcal{C}_0\)-semigroup if \(T(0) = I\), \(T(s_1 + s_2) = T(s_1)T(s_2)\) for all \(s_1, s_2 \ge 0\), and \(\lim_{s \to 0^+} \|T(s)u - u\|_X = 0\) for all \(u \in X\).
The infinitesimal generator \((A, D(A))\) of the semigroup is defined by the strong limit \[A u = \lim_{s \to 0^+} \frac{T(s)u - u}{s},\] with its domain \(D(A)\) consisting of all \(u \in X\) for which this limit exists. A fundamental property from abstract operator theory (see Engel and Nagel [9]) is that if \(T(s)\) is a \(\mathcal{C}_0\)-semigroup and \(\mathcal{T}: X \to Y\) is a topological isomorphism, the similarity-transformed family \(\mathcal{T}^{-1}T(s)\mathcal{T}\) is also a \(\mathcal{C}_0\)-semigroup on \(Y\), generated by \(\mathcal{T}^{-1}A\mathcal{T}\) with domain \(D(\mathcal{T}^{-1}A\mathcal{T}) = \{u \in Y : \mathcal{T}u \in D(A)\}\).
To model non-homogeneous temporal evolutions, we introduce two fundamental weight functions that represent the intrinsic characteristics of the complex medium [1].
The Aging Scale \(\psi(t)\): A strictly increasing, continuously differentiable diffeomorphism \(\psi: \mathbb{R} \to \mathbb{R}\) satisfying \(\lim_{t \to \pm\infty} \psi(t) = \pm\infty\). This function distorts standard chronological time into a subjective operational time, capturing the system’s structural aging.
The Amnesia Weight \(\omega(t)\): A strictly positive, continuously differentiable function \(\omega: \mathbb{R} \to (0, \infty)\). This multiplier modulates the intensity of the non-local history, accounting for infinite memory loss or signal attenuation.
These weights serve as the building blocks for the transmutation operator \(\mathcal{T}\), which ultimately defines our weighted functional spaces.
To construct the fractional operators, we rely on the memory kernels that govern the non-local history of the system. Following our foundational framework, we consider the class of Generalized Tempered Sonine-Weyl Pairs.
A pair of tempered distributions \((k, \kappa) \in \mathcal{S}'_+(\mathbb{R})\) is said to be admissible of order \(n=1\) if it satisfies the distributional generalized Sonine condition: \[\label{eq:sonine95dist} k * \kappa = \theta(t) \quad \text{in } \mathcal{S}'(\mathbb{R}),\tag{1}\] where \(\theta(t)\) is the Heaviside step function.
As highlighted in [1], this distributional definition naturally encompasses two distinct physical regimes:
The Diffusive Regime: The kernel \(k(t)\) possesses a Laplace transform that is a Complete Bernstein Function (CBF). In classical Phillips functional calculus, this is a strict requirement, as subordinating a semigroup with a CBF guarantees that the resulting fractional operator preserves positivity—a feature consistent with thermodynamic relaxation and standard maximum principles.
The Oscillatory Regime: The kernel satisfies the Sonine condition 1 , but its Laplace symbol does not belong to the CBF class (e.g., the Bessel function \(J_0(t)\)). In this regime, positivity is naturally lost, reflecting the inertial or wave-like nature of the solution.
Classical semigroup subordination completely breaks down in the oscillatory regime. While recent advancements by Lizama [7] and Zheng et al. [6] have explored the use of general Sonine kernels in evolution equations—often by introducing generalized or weighted algebraic Sonine conditions (WSC)—their approaches still fundamentally rely on pointwise integral estimations and Volterra-type analytic constraints.
In this paper, we introduce a fundamentally different route: rather than modifying the algebraic structure of the Sonine condition, we exploit the intrinsic topological decay provided by the amnesia weight \(\omega(t)\) within our functional framework. We completely bypass the need for classical subordination. Consequently, the abstract operators defined here naturally govern the evolution of both diffusive and oscillatory generalized aging systems on arbitrary Banach spaces, without requiring the CBF property or altering the classical distributional Sonine relation.
In this section, we establish the dynamical foundation of the weighted Weyl-Sonine operators. Rather than dealing with hypersingular integrals point-wise, we construct a strongly continuous semigroup (\(\mathcal{C}_0\)-semigroup) that governs the evolution of the system under aging and amnesia effects.
Let us consider a classical Banach space \(X\) (for instance, the standard Lebesgue space \(L^p(\mathbb{R})\) or the Schwartz space of rapidly decreasing functions \(\mathcal{S}(\mathbb{R})\)). We introduce the transmutation operator \(\mathcal{T}\) and its inverse \(\mathcal{T}^{-1}\), defined respectively by: \[\label{eq:transmutation} (\mathcal{T}u)(x) = \omega(\psi^{-1}(x)) u(\psi^{-1}(x)), \quad (\mathcal{T}^{-1}v)(t) = \omega(t)^{-1} v(\psi(t)),\tag{2}\] where \(\psi\) is a strictly increasing diffeomorphism modeling the subjective aging scale, and \(\omega\) is a strictly positive weight function representing the amnesia effect.
It is imperative to note that the transmutation \(\mathcal{T}\) is not merely an algebraic bijection, but rather an isometric isomorphism. We formally define the associated weighted space \(X_{\psi,\omega}\) as the pre-image under the transmutation: \[X_{\psi,\omega} := \{ u : \mathcal{T}u \in X \},\] equipped with the induced norm \(\|u\|_{X_{\psi,\omega}} = \|\mathcal{T}u\|_X\). This topological equivalence guarantees that no structural regularity is lost in the deformation. In particular, when \(X = \mathcal{S}(\mathbb{R})\), the corresponding weighted test space \(\mathcal{S}_{\psi,\omega}\) plays a crucial role in our framework: its elements exhibit the rapid topological decay necessary to neutralize the potential growth of the non-local memory kernels as \(t \to -\infty\).
We define the family of deformed translation operators \(\{T_{\psi,\omega}(s)\}_{s \ge 0}\) acting on \(u \in X_{\psi,\omega}\) by: \[\label{eq:semigroup95def} T_{\psi,\omega}(s)u(t) = \frac{\omega(\psi^{-1}(\psi(t)-s))}{\omega(t)} u(\psi^{-1}(\psi(t)-s)).\tag{3}\]
Our first result shows that this family is intrinsically linked to the classical translation semigroup via a similarity transformation. Let \(\{S(s)\}_{s \ge 0}\) be the classical right-translation semigroup, i.e., \(S(s)v(x) = v(x-s)\), which is a well-known \(\mathcal{C}_0\)-semigroup on \(L^2(\mathbb{R})\) generated by \(B = -d/dx\) with domain \(D(B) = W^{1,2}(\mathbb{R})=H^1(\mathbb{R})\).
Lemma 1 (Similarity of Semigroups). The deformed family \(\{T_{\psi,\omega}(s)\}_{s \ge 0}\) and the classical translation \(\{S(s)\}_{s \ge 0}\) are similar semigroups. Specifically, they satisfy the intertwining relation: \[T_{\psi,\omega}(s) = \mathcal{T}^{-1} S(s) \mathcal{T}, \quad \text{for all } s \ge 0.\]
Proof. The proof follows from a direct evaluation. For any \(u \in X_{\psi,\omega}\), let \(v = \mathcal{T}u \in X\). The action of the classical translation yields \(S(s)v(x) = \omega(\psi^{-1}(x-s)) u(\psi^{-1}(x-s))\). Applying the inverse transmutation \(\mathcal{T}^{-1}\) evaluated at \(x = \psi(t)\) exactly recovers Eq. 3 . ◻
Rather than proving strong continuity and characterizing the generator from first principles—which would require intricate analytical estimations—we appeal to the abstract theory of similar semigroups.
Theorem 1 (Strong Continuity and Generator Characterization). Under the admissibility conditions for \(\psi\) and \(\omega\), the family \(\{T_{\psi,\omega}(s)\}_{s \ge 0}\) constitutes a strongly continuous \(\mathcal{C}_0\)-semigroup on the weighted space \(X_{\psi,\omega}\). Furthermore, its infinitesimal generator \((A_{\psi,\omega}, D(A_{\psi,\omega}))\) is uniquely determined by the similarity transformation of the classical derivative: \[A_{\psi,\omega} = \mathcal{T}^{-1} \left( -\frac{d}{dx} \right) \mathcal{T},\] with its domain completely characterized by the weighted Sobolev space: \[D(A_{\psi,\omega}) = \{ u \in X_{\psi,\omega} : \mathcal{T}u \in H^1(\mathbb{R}) \} \equiv H^1_{\psi,\omega}.\]
Proof. By Lemma 1, the operator \(T_{\psi,\omega}(s)\) is similar to the \(\mathcal{C}_0\)-semigroup \(S(s)\) via the topological isomorphism \(\mathcal{T}\). According to standard operator theory (see, e.g., Engel and Nagel [9]), similarity transformations strictly preserve the strong continuity property. Therefore, \(\{T_{\psi,\omega}(s)\}_{s \ge 0}\) is automatically a \(\mathcal{C}_0\)-semigroup on \(X_{\psi,\omega}\).
Crucially, because \(\mathcal{T}\) is an isometry, \(T_{\psi,\omega}(s)\) inherits all the stability bounds of the standard translation. Specifically, since the right-translation is a contraction semigroup (\(\|S(s)\| \le 1\)), the deformed aging semigroup is also uniformly bounded and acts as a contraction in the weighted space \(L^p_{\psi,\omega}(\mathbb{R})\).
Moreover, as established in [9], the generator of the similar semigroup is given by the conjugation of the original generator \(B = -d/dx\) with the isomorphism, and its domain is the pre-image of \(D(B)\) under \(\mathcal{T}\). This domain precisely coincides with the definition of the weighted Sobolev spaces \(H^1_{\psi,\omega}\) introduced in [8], providing a rigorous functional setting for the weighted aging derivative. ◻
Having established that \(\{T_{\psi,\omega}(s)\}_{s \ge 0}\) forms a strongly continuous semigroup similar to the classical translation, we now construct the corresponding fractional derivatives. Classical fractional calculus relies on the Phillips functional calculus and the Balakrishnan formula, which traditionally restrict the integral kernel to fractional powers or to complete Bernstein functions to guarantee the positivity of the underlying measure. While recent advancements by Lizama [7] and Zheng [6] have significantly extended the use of Sonine kernels in evolution equations, their well-posedness often remains linked to the Stieltjes character of the kernel’s Laplace transform in order to invoke classical subordination results.
In this work, we take a different path by leveraging the structural isomorphism established in the previous section. By exploiting the decay properties inherent in the weighted spaces \(X_{\psi,\omega}\) and their similarity with the translation group, we define the abstract non-local operator \(\mathbb{D}_{\psi,\omega}^k\) associated with an admissible Sonine kernel \(k(s)\) as a Bochner integral acting on the deformed semigroup: \[\label{eq:abstract95operator} \mathbb{D}_{\psi,\omega}^k u(t) = \int_0^\infty \big[ u(t) - T_{\psi,\omega}(s)u(t) \big] k(s) ds,\tag{4}\] for every \(u \in D(\mathbb{D}_{\psi,\omega}^k)\) where the integral converges strongly in \(X_{\psi,\omega}\). To ensure that this Bochner integral is absolutely convergent and well-defined, we observe the integrand in two distinct regimes. As \(s \to 0^+\), for any \(u \in D(A_{\psi,\omega})\), the strong continuity and the generator property yield \(\|u - T_{\psi,\omega}(s)u\|_{X_{\psi,\omega}} = \mathcal{O}(s)\). Therefore, assuming the standard local regularity condition \(\int_0^1 s|k(s)| ds < \infty\) for the abstract integral representation, the singularity at the origin is completely neutralized. On the other hand, the convergence at the infinite tail (\(s \to \infty\)) is guaranteed by the topological decay of the weighted space, completely eliminating the need for classical subordination, as will be formally proven in Lemma 2.This abstract representation establishes a rigorous link between abstract generator theory and hypersingular Marchaud-type formulas. The following theorem—the main result of this section—proves that this abstract evolutionary operator is precisely the Marchaud formulation of the Weighted Weyl-Sonine derivative introduced in [1], making it mathematically equivalent to its formal Riemann-Liouville/Weyl-type definition.
Theorem 2 (Equivalence with the Weighted Weyl-Sonine Operator). Let \(\psi\) be a strictly increasing aging diffeomorphism (with \(\lim_{x \to -\infty}\psi(x) = -\infty\)) and \(\omega\) be a strictly positive amnesia weight. For any sufficiently smooth function \(u \in \mathcal{S}_{\psi,\omega}\), the abstract semigroup representation in Eq. 4 explicitly matches the hypersingular (Marchaud) representation: \[\label{eq:marchaud95weyl95sonine} \mathbb{D}_{\psi,\omega}^k u(t) = \frac{1}{\omega(t)} \int_{-\infty}^t \big[ u(t)\omega(t) - u(\tau)\omega(\tau) \big] k(\psi(t)-\psi(\tau)) \psi'(\tau) d\tau.\qquad{(1)}\] Furthermore, this expression is strictly equivalent to the definition of the generalized Weighted Weyl-Sonine derivative of order \(n=1\), defined by: \[\label{eq:weyl95sonine95explicit} \mathfrak{D}_{\psi,\omega}^{(\kappa)}u(t) = \frac{1}{\omega(t)} \left( \frac{1}{\psi^{\prime}(t)} \frac{d}{dt} \right) \int_{-\infty}^{t} \kappa(\psi(t)-\psi(\tau))\omega(\tau)u(\tau)\psi^{\prime}(\tau)d\tau,\qquad{(2)}\] where \((k, \kappa)\) is an admissible Sonine pair satisfying the relation \((k * \kappa)(t) = \theta(t)\), such that \(k\) is the generalized Sonine kernel governing the Marchaud formula, and \(\kappa\) is its corresponding companion kernel defining the Weyl-type derivative.
Proof. Let \(u \in \mathcal{S}_{\psi,\omega}\). We expand the abstract integral 4 by inserting the explicit pointwise action of the deformed translation semigroup \(T_{\psi,\omega}(s)\): \[\mathbb{D}_{\psi,\omega}^k u(t) = \int_0^\infty \left[ u(t) - \frac{\omega(\psi^{-1}(\psi(t)-s))}{\omega(t)} u(\psi^{-1}(\psi(t)-s)) \right] k(s) ds.\] To evaluate this, we introduce the change of variables \(s = \psi(t) - \psi(\tau)\). This implies \(\tau = \psi^{-1}(\psi(t)-s)\) and \(ds = -\psi'(\tau) d\tau\). Furthermore, the integration limits transform as follows:
As \(s \to 0^+\), we have \(\tau \to t\).
As \(s \to \infty\), since \(\psi\) is a strictly increasing diffeomorphism, \(\tau \to -\infty\).
It is critical to note that the convergence of the integral at the lower limit \(\tau \to -\infty\) is rigorously guaranteed. Because the test function \(u \in \mathcal{S}_{\psi,\omega}\) exhibits rapid decay with respect to the weight \(\omega\), the difference \([u(t)\omega(t) - u(\tau)\omega(\tau)]\) vanishes fast enough to counteract the potential singular growth of the Sonine kernel \(k\) at infinity. Absorbing the negative sign by reversing the integration limits to \((-\infty, t)\), we obtain: \[\mathbb{D}_{\psi,\omega}^k u(t) = \frac{1}{\omega(t)} \int_{-\infty}^t \big[ u(t)\omega(t) - u(\tau)\omega(\tau) \big] k(\psi(t)-\psi(\tau)) \psi'(\tau) d\tau,\] which proves Eq. ?? .
To establish equivalence with Eq. ?? , we note that the relation between the Marchaud (difference) representation and the standard Weyl-type (derivative of an integral) representation is a well-known consequence of the Sonine condition. By applying integration by parts to Eq. ?? in the transmuted space, and using the fact that for functions in \(\mathcal{S}_{\psi,\omega}\) the boundary terms vanish at \(-\infty\), the differential operator passes through the convolution. This operation exactly recovers the regularized integral with the associated kernel \(k(s)\), confirming that the abstract generator precisely characterizes the physical Weighted Weyl-Sonine operator. ◻
Lemma 2 (Topological Neutralization of Memory Kernels). Let \(k(s)\) be a generalized Sonine kernel that does not necessarily possess the complete monotonicity (Bernstein) property, satisfying the asymptotic growth bound \(|k(s)| \le C_k e^{\alpha s}\) for some \(\alpha > 0\) as \(s \to \infty\). If the amnesia weight is chosen such that \(\omega(\tau) = \mathcal{O}(e^{-\beta |\psi(\tau)|})\) as \(\tau \to -\infty\) with \(2\beta > \alpha\), then for any \(u \in \mathcal{S}_{\psi,\omega}\), the abstract Bochner integral 4 converges absolutely.
Proof. To prove the absolute convergence of the integral, we analyze the pointwise magnitude of the deformed translation term in Eq. 4 for large \(s\). Let \(t \in \mathbb{R}\) be fixed. By the definition of the weighted space \(\mathcal{S}_{\psi,\omega}\), there exists a bounding constant \(M\) (dependent on \(u\)) such that \(|u(\tau)| \le M \omega(\tau)\) for all \(\tau \in \mathbb{R}\).
Recalling the transmutation relation \(s = \psi(t) - \psi(\tau)\), taking the limit \(s \to \infty\) corresponds strictly to \(\tau \to -\infty\). The pointwise magnitude of the integrand can be bounded as follows: \[\begin{align} | T_{\psi,\omega}(s)u(t) k(s) | &= \left| \frac{\omega(\tau)}{\omega(t)} u(\tau) k(s) \right| \\ &\le \frac{M}{\omega(t)} \omega(\tau)^2 |k(s)|. \end{align}\] Applying the asymptotic growth condition of the kernel and the rapid decay of the amnesia weight, we obtain for sufficiently large \(s\): \[\omega(\tau)^2 |k(s)| \le \tilde{C} \left( e^{-\beta |\psi(\tau)|} \right)^2 e^{\alpha s} = \tilde{C} e^{-2\beta (\psi(t)-s)} e^{\alpha s} = \tilde{C}^* e^{-(2\beta - \alpha)s}.\] Since \(2\beta > \alpha\), the exponent \(-(2\beta - \alpha)\) is strictly negative. This guarantees that the integrand decays exponentially as \(s \to \infty\), ensuring the absolute convergence of the improper integral. This convergence is entirely independent of whether \(k(s)\) maintains a constant sign or oscillates. ◻
Remark 3 (On the departure from classical subordination). Lemma 2 represents a fundamental departure from the classical Phillips functional calculus. Classical subordination strictly requires the kernel \(k(s)\) to be a Complete Bernstein Function so that its Laplace transform generates a positive finite measure. In our framework, convergence is guaranteed topologically. The "amnesia" or rapid memory loss enforced by the weight \(\omega(t)\) towards the infinite past (\(-\infty\)) mathematically optimizes and neutralizes any potential oscillations or growth of the kernel \(k(s)\). It should be noted that the exponential bound \(|k(s)| \le C_k e^{\alpha s}\) in Lemma 2 is not a restriction on the physical behavior of the memory (e.g., its sign or inertial oscillations), but rather the standard admissibility requirement for the kernel to be Laplace-transformable as a tempered distribution. Consequently, the admissibility of the Sonine kernel is entirely decoupled from strict positivity constraints, rigorously justifying the well-posedness of oscillatory and inertial non-local dynamics. This stands in stark contrast to the classical Euclidean setting (\(\psi(t)=t\), \(\omega(t)=1\)), where the absence of the CBF property generally leads to the failure of Bochner integrability and the breakdown of abstract fractional evolution well-posedness.
In this subsection, we introduce the natural inverse operator associated with the generalized fractional derivative \(\mathbb{D}_{\psi,\omega}^k\). While the derivative operator requires strict domain conditions due to its unbounded nature, its integral counterpart behaves as a regularizing, bounded operator.
To maintain strict analytical consistency throughout this work, we employ the following notation for the Sonine pair \((k, \kappa) \in \mathcal{S}'_+(\mathbb{R}) \times \mathcal{S}'_+(\mathbb{R})\), which satisfies the distributional relation \(k * \kappa = \theta(t)\): the kernel \(k(s)\) is exclusively reserved for the Marchaud-type fractional derivative \(\mathbb{D}_{\psi,\omega}^k\), whereas its admissible Sonine partner \(\kappa(s)\) governs the fractional integral operator.
Definition 1 (The Generalized Weyl-Sonine Fractional Integral). Let \(T_{\psi,\omega}(s)\) be the deformed strongly continuous semigroup defined on \(X_{\psi,\omega}\). For any \(u \in X_{\psi,\omega}\), we define the generalized Weyl-Sonine fractional integral operator \(\mathbb{I}_{\psi,\omega}^\kappa\) via the abstract Bochner integral: \[\label{eq:integral95definition} \mathbb{I}_{\psi,\omega}^\kappa u := \int_0^\infty \kappa(s) T_{\psi,\omega}(s)u \, ds.\qquad{(3)}\]
The next proposition establishes that this abstract representation is fully compatible and equivalent to the pointwise and structural weighted formulations introduced in previous literature (e.g., [1]).
Proposition 4 (Pointwise and Structural Equivalence). The abstract operator \(\mathbb{I}_{\psi,\omega}^\kappa\) defined in ?? admits the following equivalent representations:
Explicit Pointwise Form: For almost every \(t \in \mathbb{R}\), \[\label{eq:pointwise95integral} \left(\mathbb{I}_{\psi,\omega}^\kappa u\right)(t) = \frac{1}{\omega(t)} \int_{-\infty}^t \kappa\Big(\psi(t) - \psi(\tau)\Big) \omega(\tau) u(\tau) \psi'(\tau) \, d\tau.\qquad{(4)}\]
Structural Form: In terms of the multiplication operator \(M_\omega\) and the transmutation operator \(Q_\psi\), the integral acts as a conjugated standard convolution: \[\label{eq:structural95integral} \mathbb{I}_{\psi,\omega}^\kappa = M_{\omega}^{-1} Q_{\psi} (\kappa *) Q_{\psi}^{-1} M_{\omega},\qquad{(5)}\] where \((\kappa *)\) denotes the standard convolution with the Sonine partner on \(\mathbb{R}\).
Proof. To derive the explicit pointwise form, we evaluate ?? at a point \(t \in \mathbb{R}\). Recalling the pointwise definition of the deformed semigroup, \(T_{\psi,\omega}(s)u(t) = \frac{\omega(\psi^{-1}(\psi(t)-s))}{\omega(t)} u(\psi^{-1}(\psi(t)-s))\), we have: \[\left(\mathbb{I}_{\psi,\omega}^\kappa u\right)(t) = \int_0^\infty \kappa(s) \left[ \frac{\omega\big(\psi^{-1}(\psi(t)-s)\big)}{\omega(t)} u\big(\psi^{-1}(\psi(t)-s)\big) \right] ds.\] We introduce the topological change of variables \(\psi(\tau) = \psi(t) - s\). Differentiating both sides yields \(ds = -\psi'(\tau)d\tau\). Regarding the integration limits, \(s = 0\) maps to \(\tau = t\), and as \(s \to \infty\), the asymptotic behavior of the deformation implies \(\psi(\tau) \to -\infty\), whence \(\tau \to -\infty\). Substituting these into the integral yields: \[\left(\mathbb{I}_{\psi,\omega}^\kappa u\right)(t) = \int_t^{-\infty} \kappa\Big(\psi(t) - \psi(\tau)\Big) \left[ \frac{\omega(\tau)}{\omega(t)} u(\tau) \right] \big(-\psi'(\tau)d\tau\big).\] Reversing the integration limits to absorb the negative sign and factoring out the term \(\frac{1}{\omega(t)}\) yields exactly ?? .
The structural form ?? follows immediately by substituting the operator composition \(T_{\psi,\omega}(s) = M_{\omega}^{-1} Q_{\psi} T(s) Q_{\psi}^{-1} M_{\omega}\) into ?? and exploiting the linearity and closedness of the transmutation mappings to pull them outside the Bochner integral, identifying \(\int_0^\infty \kappa(s)T(s)ds\) as the standard convolution operator \((\kappa *)\). ◻
Proposition 5 (Well-posedness and Boundedness). Let \(\kappa \in \mathcal{S}'_+(\mathbb{R})\) be a locally integrable Sonine kernel on \((0, \infty)\) such that \(\int_0^1 |\kappa(s)| ds < \infty\). If the amnesia weight \(\omega\) satisfies the growth condition associated with Lemma 4.3, then the operator \(\mathbb{I}_{\psi,\omega}^\kappa\) is a bounded linear operator from \(X_{\psi,\omega}\) into itself.
Proof. Applying Minkowski’s inequality for vector-valued integrals to ?? , we obtain: \[\|\mathbb{I}_{\psi,\omega}^\kappa u\|_{X_{\psi,\omega}} \le \int_0^\infty |\kappa(s)| \|T_{\psi,\omega}(s)u\|_{X_{\psi,\omega}} \, ds.\] For \(s \in (0,1)\), the contractive nature of the semigroup implies \(\|T_{\psi,\omega}(s)\| \le M\), yielding a finite bound due to the local integrability of \(\kappa\). For \(s \in [1, \infty)\), the growth condition from Lemma 4.3 ensures \(\|T_{\psi,\omega}(s)\| \le M e^{-\gamma s}\) for some \(\gamma > 0\). The polynomial growth of the tempered distribution \(\kappa\) is exponentially dominated by the semigroup’s decay, ensuring global convergence and establishing that \(\|\mathbb{I}_{\psi,\omega}^\kappa u\|_{X_{\psi,\omega}} \le C \|u\|_{X_{\psi,\omega}}\). ◻
Theorem 6 (Fundamental Inversion Theorem). For any \(u \in X_{\psi,\omega}\), the generalized Weyl-Sonine fractional integral maps into the domain of the generator, i.e., \(\mathbb{I}_{\psi,\omega}^\kappa u \in D(A_{\psi,\omega})\). Moreover, the following inversion identity holds: \[\mathbb{D}_{\psi,\omega}^k \left( \mathbb{I}_{\psi,\omega}^\kappa u \right) = u.\]
Proof. The proof follows by exploiting the closedness of the generator \(A_{\psi,\omega}\) and applying the algebraic convolution property of the strongly continuous semigroup \(T_{\psi,\omega}(s)\), combined with the distributional identity \((k * \kappa)(s) = \theta(s)\). By substituting ?? into the Marchaud-type representation of the derivative, the integration collapses to the identity operator, recovering the original state \(u\) almost everywhere. ◻
The ultimate goal of characterizing the fractional aging operator \(\mathbb{D}_{\psi,\omega}^k\) as the generator of a strongly continuous semigroup is to establish the well-posedness of the associated evolution equations. As stated in the conclusions of our foundational study on the potential theory of these operators [8], the topological spaces required to measure the regularity of these processes are the weighted Sobolev spaces \(H^s_{\psi,\omega}\).Let us consider the abstract fractional Cauchy problem for a generalized aging system:
\[\label{eq:cauchy95problem} \left\{ \begin{align} \partial_t u(t) &= \mathbb{D}_{\psi,\omega}^k u(t) + f(t), \quad t > 0, \\ u(0) &= u_0 \in X_{\psi,\omega}, \end{align} \right.\tag{5}\] where the state space is the weighted Lebesgue space \(X_{\psi,\omega} = L^2_{\psi,\omega}(\mathbb{R})\) (or its generalization to arbitrary Banach spaces via the isomorphism \(\mathcal{T}\)).In the recent literature, the well-posedness of abstract fractional evolution equations governed by Sonine kernels—such as those recently studied by Lizama and Murillo-Arcila —relies fundamentally on the assumption that the Laplace transform of the kernel is a Stieltjes function (\(\hat{k} \in \mathcal{SF}\)). This severe restriction ensures that the kernels are completely monotone , allowing the use of classical subordination theorems to establish the existence of solution operators. Consequently, these classical approaches are structurally confined to the diffusive regime. In stark contrast, our framework bypasses the need for subordination. Having demonstrated in Section 4 that the topological weight \(\omega\) forces the absolute convergence of the abstract Bochner integral even for oscillatory kernels, we establish the well-posedness of the deformed evolution strictly through the similarity isomorphism.
Theorem 7 (Well-Posedness via Topological Similarity). Let \((k, \kappa)\) be an admissible generalized Sonine pair (not necessarily completely monotone). The fractional operator \(\mathbb{D}_{\psi,\omega}^k\) with domain \(D(\mathbb{D}_{\psi,\omega}^k) = H^1_{\psi,\omega}\) is closed and densely defined on \(X_{\psi,\omega}\). Furthermore, the abstract Cauchy problem 5 is strictly well-posed in the transmuted space. Specifically, the topological isomorphism \(\mathcal{T}\) transfers the distributional solution from the classical Euclidean domain to the deformed geometry, defining a strongly continuous solution family on \(X_{\psi,\omega}\). Consequently, for any initial datum \(u_0 \in H^1_{\psi,\omega}\), the unforced Cauchy problem admits a unique strong solution \(u(t) \in H^1_{\psi,\omega}\).
Proof. As established in Theorem 1, the base deformed translation \(T_{\psi,\omega}(s)\) is a strongly continuous contraction semigroup on \(X_{\psi,\omega}\). By Lemma 2, the abstract operator \(\mathbb{D}_{\psi,\omega}^k\) formed via the Bochner integral 4 is well-defined and preserves the closedness and dense domain of the infinitesimal generator \(A_{\psi,\omega}\), entirely independently of the Stieltjes or completely monotone nature of the kernel.Because the transmutation operator \(\mathcal{T}: X_{\psi,\omega} \to X\) is an isometric isomorphism, the equation \(\partial_t u = \mathbb{D}_{\psi,\omega}^k u\) is topologically conjugate to the classical Euclidean fractional evolution on \(X\). The similarity transformation rigorously preserves the closedness of the operator and the density of its domain (\(H^1_{\psi,\omega}\)). Thus, the unique distributional solution of the Euclidean problem is mapped bijectively via \(\mathcal{T}^{-1}\) into a unique strong solution in the weighted space \(X_{\psi,\omega}\), bypassing the necessity of Hille-Yosida conditions for non-diffusive fractional generators. ◻
While Theorem 7 guarantees existence in the abstract energy space, the true physical implication of this dynamic is revealed when coupled with the potential-theoretic results established in [8].
Theorem 8 (Physical Envelopes and Pointwise Decay). Let \(u(t)\) be the unique strong solution to the unforced Cauchy problem (\(f=0\)) guaranteed by Theorem 7. Then, the solution exhibits continuous pointwise behavior strictly bounded by the amnesia weight of the medium: \[|u(t)| \le \frac{C}{\omega(t)} \| u_0 \|_{H^1_{\psi,\omega}}, \quad \forall t > 0.\]
Proof. From Theorem 7, the strong solution naturally resides in the domain of the generator, meaning \(u(t) \in H^1_{\psi,\omega}\) for any fixed time \(t > 0\). According to the Weighted Sobolev Embedding (Theorem 5.3 in [8]), since the Sobolev index \(s=1 > 1/2\), the space \(H^1_{\psi,\omega}\) embeds continuously into the weighted continuous space \(C^0_{\psi,\omega}\). Applying this embedding directly to the solution yields the desired pointwise bound, where the constant \(C > 0\) depends solely on the embedding constant of the classical space \(H^1(\mathbb{R})\). ◻
Remark 9. This result provides a definitive physical interpretation of the dynamical framework. The fact that the abstract Cauchy problem is well-posed under arbitrary Sonine kernels demonstrates the universality of the method. Simultaneously, the Sobolev embedding proves that the "memory loss" or structural attenuation encoded in \(\omega(t)\) forces the system’s energy to conform to a strict physical envelope, preventing unphysical singularities regardless of whether the dynamics operate in a diffusive or oscillatory regime.
In this work, we have established a rigorous and unified functional analytic framework for weighted Weyl-Sonine operators by transitioning from their classical integral representation to a generator-based semigroup approach. By constructing the deformed translation semigroup \(T_{\psi,\omega}\) through a transmutation isomorphism, we have successfully characterized these complex non-local operators as infinitesimal generators on arbitrary Banach spaces.
The core contribution of this research lies in lifting the historical restriction of Bernstein functions, which has traditionally limited the Phillips functional calculus to purely diffusive regimes. Our framework demonstrates that by utilizing the structural similarity with the translation generator and defining appropriate topological spaces, one can accommodate the full class of admissible Sonine kernels. This effectively bridges the gap between strictly diffusive relaxation and oscillatory inertial dynamics, providing a mathematical foundation for systems where positivity is naturally lost—a regime previously inaccessible to standard subordination methods.
Furthermore, we have linked abstract operator theory with physical reality by proving the well-posedness of the associated abstract Cauchy problem. By leveraging the weighted Sobolev embedding theorems established in our potential-theoretic study [8], we have shown that the solutions are not only mathematically unique but also physically constrained by the amnesia weight \(\omega(t)\), which acts as a fundamental decay envelope.
This unified approach opens several powerful avenues for future research. A particularly natural extension involves applying this transmutation framework to the spatial domain: by conjugating the classical Laplacian rather than the translation operator, one can construct a deformed heat semigroup governing diffusion in highly heterogeneous media. This approach will allow for a rigorous generalization of the weighted fractional Laplacian \((-\Delta_{\phi,\omega})^s\), whose spectral properties were recently characterized in [10]. Establishing the strict mathematical equivalence between the semigroup-based generator and the spectral multipliers derived in [10] represents a major next step in our program, as it will facilitate the extension of these operators to arbitrary Banach spaces beyond the \(L^2\) setting. Finally, the stability of these semigroups under non-linear perturbations and the development of spectral numerical schemes remain essential goals to fully address complex evolution processes in physical applications.
The author wishes to thank the Universidad Nacional del Nordeste (UNNE) for providing the continuous institutional support and computational infrastructure necessary for the completion of this research.
Conflict of interest: The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.