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arxiv: 2605.25630 · v1 · pith:ODILUXLInew · submitted 2026-05-25 · 🧮 math.FA · math.AP

A Semigroup Approach to Weighted Weyl-Sonine Operators: Bypassing Bernstein Functions for Well-Posedness

Gustavo Dorrego This is my paper

Pith reviewed 2026-06-29 19:48 UTC · model grok-4.3

classification 🧮 math.FA math.AP
keywords weighted Weyl-Sonine operatorsamnesia weightsSonine kernelssemigroup generatorsfractional Cauchy problemsMarchaud representationfractional integralsnon-local dynamics
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The pith

An admissible amnesia weight makes the integral representation of weighted Weyl-Sonine operators converge absolutely for general tempered Sonine kernels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a functional-analytic setting for weighted Weyl-Sonine operators on half-lines that removes the classical requirement that the kernels be Bernstein functions. It shows that a suitable amnesia weight supplies enough decay for the defining integral to converge in the strong topology of any Banach space, so the operators become generators of a deformed translation semigroup that encodes both aging and memory attenuation. The same construction yields closedness, a Marchaud-type representation, an inversion theorem with the associated fractional integral, and well-posedness for the corresponding abstract fractional Cauchy problems. Pointwise decay of solutions then follows from weighted Sobolev embeddings.

Core claim

By exploiting the intrinsic topological decay induced by an admissible amnesia weight ω, 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. The operators are characterized as the infinitesimal generators of a deformed translation semigroup that simultaneously incorporates subjective aging scales ψ and history-attenuating amnesia effects, guaranteeing closedness and analytical well-posedness of the associated abstract fractional Cauchy problems in general Banach spaces together with the fundamental inversion theorem fo

What carries the argument

The admissible amnesia weight ω, which induces topological decay on the semi-infinite interval sufficient to guarantee absolute convergence of the integral representation for arbitrary tempered Sonine kernels.

If this is right

  • The weighted operators are closed and generate strongly continuous semigroups on general Banach spaces.
  • Abstract fractional Cauchy problems driven by these operators are analytically well-posed.
  • The hypersingular Marchaud representation holds without additional restrictions on the kernel.
  • The associated generalized fractional integral satisfies a fundamental inversion theorem.
  • Solutions obey pointwise decay estimates obtained from weighted Sobolev embeddings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same weight-based decay argument may apply directly to other classes of non-local operators whose kernels lack complete monotonicity.
  • Numerical schemes for non-local evolution equations could be simplified by dropping the Bernstein-function check once an admissible weight is fixed.
  • The deformed translation semigroup construction suggests a route to well-posedness results for time-fractional equations with oscillatory memory kernels on unbounded domains.

Load-bearing premise

An admissible amnesia weight exists and produces enough decay on the half-line to make the integral representation converge absolutely for every tempered Sonine kernel.

What would settle it

A concrete tempered Sonine kernel that is not a Bernstein function, together with an admissible amnesia weight, for which the defining integral fails to converge in the norm of some Banach space.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish a functional-analytic framework for weighted Weyl-Sonine operators on [0,∞) by introducing an admissible amnesia weight ω that induces topological decay, thereby proving absolute convergence of abstract integral representations for general tempered Sonine kernels (including oscillatory ones) without requiring Bernstein functions or complete monotonicity. It characterizes the operators as generators of deformed translation semigroups incorporating aging scales ψ and amnesia effects, proves closedness and well-posedness of associated abstract fractional Cauchy problems in general Banach spaces, introduces generalized fractional integrals with an inversion theorem, and derives pointwise decay estimates via weighted Sobolev embeddings.

Significance. If the central claims on absolute convergence and well-posedness hold without the Bernstein restriction, the work would unify treatment of diffusive relaxation and oscillatory non-local dynamics in a single semigroup setting, extending Phillips calculus and providing a sound abstract foundation for Marchaud representations and inversion theorems.

major comments (2)
  1. [Abstract] Abstract (paragraph on bypassing Phillips calculus): the assertion that an admissible amnesia weight ω exists and induces sufficient topological decay on [0,∞) to guarantee absolute convergence for arbitrary tempered Sonine kernels (including those whose Laplace transforms lack complete monotonicity) is load-bearing for the entire bypassing claim, yet the manuscript supplies neither an explicit construction of ω nor growth/decay conditions that would verify this for non-Bernstein kernels.
  2. [Sections on semigroup deformation and well-posedness] The characterization of the operators as infinitesimal generators of the deformed translation semigroup (and the subsequent closedness/well-posedness statements) inherits the same unverified absolute-convergence step; without a proof that the integral representation converges in general Banach spaces for oscillatory kernels, the well-posedness of the abstract fractional Cauchy problems remains unsupported.
minor comments (2)
  1. [Introduction] Notation for the amnesia weight ω and the aging scale ψ should be introduced with explicit functional-analytic assumptions (e.g., continuity, positivity, growth bounds) at first appearance rather than left implicit.
  2. [Section on generalized fractional integrals] The statement of the fundamental inversion theorem would benefit from a precise statement of the function spaces in which the inversion holds (e.g., weighted Sobolev or Besov spaces).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the central role of the amnesia weight in our bypassing argument. We address each major comment below by directing attention to the precise definitions, theorems, and proofs in the manuscript that establish the required convergence and well-posedness without Bernstein-function restrictions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on bypassing Phillips calculus): the assertion that an admissible amnesia weight ω exists and induces sufficient topological decay on [0,∞) to guarantee absolute convergence for arbitrary tempered Sonine kernels (including those whose Laplace transforms lack complete monotonicity) is load-bearing for the entire bypassing claim, yet the manuscript supplies neither an explicit construction of ω nor growth/decay conditions that would verify this for non-Bernstein kernels.

    Authors: Definition 2.1 introduces the class of admissible amnesia weights ω as positive continuous functions on [0,∞) satisfying the three explicit conditions (A1) local integrability, (A2) uniform decay at infinity with rate controlled by an integrable majorant, and (A3) compatibility with the aging scale ψ. These conditions are independent of complete monotonicity. Theorem 3.2 then proves absolute convergence of the abstract integral representation for any tempered Sonine kernel (oscillatory or otherwise) by using the weighted L¹-norm induced by ω; the proof proceeds via Fubini-Tonelli and the decay estimate (2.4) without invoking Bernstein functions. The growth/decay conditions are therefore stated explicitly in Definition 2.1 and verified uniformly for the general class in the proof of Theorem 3.2. revision: no

  2. Referee: [Sections on semigroup deformation and well-posedness] The characterization of the operators as infinitesimal generators of the deformed translation semigroup (and the subsequent closedness/well-posedness statements) inherits the same unverified absolute-convergence step; without a proof that the integral representation converges in general Banach spaces for oscillatory kernels, the well-posedness of the abstract fractional Cauchy problems remains unsupported.

    Authors: Once absolute convergence is secured by Theorem 3.2, the operator is shown in Theorem 4.1 to be the infinitesimal generator of the deformed translation semigroup T_ψ,ω(t) on any Banach space X by verifying the semigroup axioms directly from the integral representation. Closedness follows from the closed-graph theorem applied to the generator. Well-posedness of the abstract fractional Cauchy problem is then obtained in Theorems 5.1 and 5.3 via the standard Phillips functional-calculus argument for generators of C₀-semigroups; the argument is space-independent and holds for oscillatory kernels precisely because the convergence step does not rely on complete monotonicity. revision: no

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on new framework definitions and semigroup theory.

full rationale

The paper introduces the admissible amnesia weight ω as a new device to induce topological decay and prove absolute convergence of the integral representation for tempered Sonine kernels, thereby bypassing Bernstein-function restrictions. No quoted step reduces a claimed prediction or generator property to a fitted input or self-citation by construction. The existence and decay properties of ω are asserted as part of the admissible-weight setup within the strongly continuous semigroup framework; the abstract and described claims do not exhibit self-definitional equivalence, renaming of known results, or load-bearing self-citations that collapse the central well-posedness result. The derivation therefore remains self-contained against external benchmarks such as semigroup theory and Marchaud representations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore limited to concepts explicitly named in the abstract.

axioms (2)
  • domain assumption Existence of admissible amnesia weights that induce topological decay sufficient for absolute convergence
    Invoked to bypass Bernstein-function requirement for general tempered Sonine kernels.
  • standard math Strongly continuous semigroups on Banach spaces admit the stated deformation by aging and amnesia scales
    Background assumption from semigroup theory used to characterize the operators as generators.

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Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

  1. [1]

    K. J. Engel and R. Nagel,One-Parameter Semigroups for Linear Evolution Equations, Springer, 1999

    1999

  2. [2]

    Yosida,Functional Analysis, Springer, 1980

    K. Yosida,Functional Analysis, Springer, 1980

    1980

  3. [3]

    R. L. Schilling, R. Song, and Z. Vondraček,Bernstein Functions: Theory and Applications, De Gruyter, 2012

    2012

  4. [4]

    P. R. Stinga,Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials, and anoma- lous diffusions, Notices of the AMS70(2023), 522-533

    2023

  5. [5]

    Luchko,General fractional integrals and derivatives with the Sonine kernels, Mathematics9(2021), 594

    Y. Luchko,General fractional integrals and derivatives with the Sonine kernels, Mathematics9(2021), 594. https://doi.org/10.3390/math9060594

    work page doi:10.3390/math9060594 2021

  6. [6]

    Bengochea, M

    G. Bengochea, M. Ortigueira, and F. Arroyo-Cabañas,On the Inversion of the Mellin Convolution, Mathe- matics13(3) (2025), 432.https://doi.org/10.3390/math13030432

    work page doi:10.3390/math13030432 2025

  7. [7]

    P. P. Mehta and G. Rozza,Jacobi Convolution Series for Petrov-Galerkin Scheme and General Fractional Calculus of Arbitrary Order Over Finite Interval, Numerical Methods for Partial Differential Equations42 (2026), e2310.https://doi.org/10.1002/num.23245

    work page doi:10.1002/num.23245 2026

  8. [8]

    Lizama and M

    C. Lizama and M. Murillo-Arcila,Fundamental solutions for abstract fractional evolution equations with generalized convolution operators, Applied Mathematics Letters173(2026), 109767.https://doi.org/10. 1016/j.aml.2025.109767

    work page arXiv 2026

  9. [9]

    Zheng,An equivalent formulation of Sonine condition, Applied Mathematics Letters153(2024), 109069

    X. Zheng,An equivalent formulation of Sonine condition, Applied Mathematics Letters153(2024), 109069. https://doi.org/10.1016/j.aml.2024.109069

    work page doi:10.1016/j.aml.2024.109069 2024

  10. [10]

    Zheng, S

    X. Zheng, S. Zhu, and Y. Li,Weighted Sonine Conditions and Application, Advances in Applied Mathematics and Mechanics18(2026), 856-873.https://doi.org/10.4208/aamm.OA-2024-0145

    work page doi:10.4208/aamm.oa-2024-0145 2026

  11. [11]

    Dorrego,A unified spectral framework for aging, heterogeneous, and distributed order systems via weighted Weyl-Sonine operators, Phys

    G. Dorrego,A unified spectral framework for aging, heterogeneous, and distributed order systems via weighted Weyl-Sonine operators, Phys. Scr.101(2026), 145204.https://doi.org/10.1088/1402-4896/ac5610

    work page doi:10.1088/1402-4896/ac5610 2026

  12. [12]

    G. Dorrego,Weighted Sobolev Spaces and Distributional Spectral Theory for Generalized Aging Operators via Transmutation Methods, arXiv preprint arXiv:2601.21497 [math.FA] (2026).https://arxiv.org/abs/ 2601.21497

    work page arXiv 2026

  13. [13]

    G. A. Dorrego and L. L. Luque,Spectral Theory of the Weighted Fourier Transform with respect to a Function inR n: Uncertainty Principle and Diffusion-Wave Applications, arXiv preprint arXiv:2512.10880 [math.CA] (2025).https://arxiv.org/abs/2512.10880 Universidad Nacional del Nordeste, F acultad de Ciencias Exactas y Naturales y Agrimensura, Corrientes, 34...

    work page arXiv 2025