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arxiv: 2603.14715 · v2 · pith:YGMVYA3Inew · submitted 2026-03-16 · 🧮 math.AP · math-ph· math.MP

Towards a Gagliardo-Type Theory of Fractional Sobolev Spaces on Arbitrary Time Scales

Hafida Abbas , Abdelhalim Azzouz , Praveen Agarwal , Delfim F. M. Torres This is my paper

Pith reviewed 2026-05-22 10:21 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords fractional Sobolev spacestime scalesGagliardo seminormnonlocal regularityPoincaré inequalitySobolev embeddinghybrid domainsdelta measure
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The pith

A Gagliardo seminorm on the delta measure defines nonlocal fractional Sobolev spaces on time scales.

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

The paper constructs fractional Sobolev spaces on arbitrary time scales by adapting the classical Gagliardo seminorm to the Lebesgue delta-measure and the off-diagonal part of the product measure. This produces a genuinely nonlocal notion of fractional regularity that does not rely on fractional derivatives. The resulting spaces are Banach for finite exponents, reflexive in the open range of exponents, and Hilbert when the exponent is two. On bounded hybrid time scales whose connected components are separated by a positive distance, the construction yields a Poincaré inequality, a fractional Sobolev embedding, and Hardy- and Caffarelli-Kohn-Nirenberg-type inequalities for subcritical weights.

Core claim

The central claim is that a Gagliardo-type seminorm, defined via the Lebesgue delta-measure and the off-diagonal interaction domain of the product measure, equips arbitrary time scales with fractional Sobolev spaces that are structurally distinct from all derivative-based constructions previously studied on these domains. On continuous intervals the new spaces are equivalent to the bilateral Riemann-Liouville spaces (restricted to functions with vanishing boundary trace) in the supercritical regime, while on hybrid scales an explicit obstruction prevents any such equivalence because of mixed continuous-discrete interactions. When the time scale is bounded and its components are separated by

What carries the argument

The Gagliardo seminorm that integrates the p-th power of the difference of function values against the product of delta-measures over the off-diagonal set induced by the time-scale product measure.

If this is right

  • The spaces are complete normed spaces for every admissible fractional order and integrability exponent.
  • Reflexivity holds whenever the integrability exponent lies strictly between one and infinity.
  • The quadratic case yields Hilbert spaces.
  • A sharp nontriviality criterion is available on any bounded time scale with finitely many components.
  • On separated hybrid scales the framework supplies Poincaré, Sobolev-embedding, Hardy, and Caffarelli-Kohn-Nirenberg inequalities for subcritical weights.

Where Pith is reading between the lines

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

  • The nonlocal character may permit direct treatment of equations with jump discontinuities or mixed continuous-discrete dynamics without first passing through a derivative operator.
  • The explicit obstruction on hybrid scales suggests that any future theory of nonlocal PDEs on time scales must retain the full product-measure interaction term rather than reduce to separate continuous and discrete parts.
  • Relaxing the positive-separation hypothesis while retaining control of the mixed terms would constitute a natural next step toward unbounded or densely interleaved hybrid domains.

Load-bearing premise

The time scale must be bounded with finitely many connected components separated by a positive distance for the Poincaré inequality, Sobolev embedding, and Hardy-type inequalities to hold.

What would settle it

A concrete counter-example on a hybrid time scale with two components whose separation distance is allowed to approach zero, in which the Poincaré constant becomes unbounded or the inequality fails outright.

read the original abstract

We propose a systematic Gagliardo-type formulation of fractional Sobolev spaces on arbitrary time scales, based on the Lebesgue Delta-measure and the off-diagonal interaction domain induced by the product measure. For fractional orders strictly between zero and one and for finite Lebesgue exponents, we define a nonlocal Gagliardo seminorm and the associated function space. This construction provides a notion of fractional regularity on time scales that is genuinely nonlocal and structurally distinct from the derivative-based approaches developed in the existing literature. We establish the basic functional properties of these spaces: they are Banach spaces in all admissible cases, reflexive in the strict range of exponents, and Hilbert in the quadratic case. On bounded time scales with finitely many connected components, we identify a sharp criterion for the construction to be nontrivial. We then compare the new framework with the derivative-based Riemann--Liouville fractional Sobolev spaces previously studied on time scales. On a continuous interval, in the supercritical regime, we obtain a norm equivalence with the bilateral Riemann--Liouville space on the subspace of functions with vanishing boundary trace. On hybrid time scales, we prove an explicit obstruction that rules out any analogous equivalence, due to the contribution of the mixed continuous--discrete interactions. On bounded hybrid time scales with finitely many connected components separated by a positive distance, we further establish a Poincar\'e-type inequality, a fractional Sobolev embedding, and fractional Hardy and Caffarelli--Kohn--Nirenberg-type inequalities for subcritical weights. Together, these results provide a complete functional and geometric framework, together with first geometric estimates, for the nonlocal Gagliardo-type approach to fractional Sobolev spaces on time scales.

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

1 major / 1 minor

Summary. The paper introduces a Gagliardo-type definition of fractional Sobolev spaces on arbitrary time scales, using the Lebesgue Δ-measure and the off-diagonal interaction domain induced by the product measure to define a nonlocal seminorm for 0 < s < 1. It proves that the resulting spaces are Banach in general, reflexive for 1 < p < ∞, and Hilbert when p=2. On continuous intervals, norm equivalence holds with the bilateral Riemann-Liouville fractional Sobolev space (in the supercritical regime) on the subspace of functions with vanishing boundary trace. On hybrid time scales an explicit obstruction to any such equivalence is shown, arising from mixed continuous-discrete contributions in the seminorm. On bounded hybrid time scales whose finitely many connected components are separated by a positive distance, the authors establish a Poincaré-type inequality, a fractional Sobolev embedding, and fractional Hardy and Caffarelli-Kohn-Nirenberg inequalities for subcritical weights.

Significance. If the geometric estimates hold, the construction supplies a genuinely nonlocal notion of fractional regularity on time scales that is structurally distinct from existing derivative-based (Riemann-Liouville) approaches. The explicit obstruction on hybrids and the first geometric inequalities under the positive-separation hypothesis constitute concrete advances that could serve as a foundation for nonlocal analysis on hybrid domains.

major comments (1)
  1. [Results on bounded hybrid time scales with positive separation (abstract and the section establishing the Poincaré-type,] The positive separation of connected components is invoked to control mixed continuous-discrete interactions in the Gagliardo seminorm (see the statement of the Poincaré, embedding, and Hardy/CKN results). However, the manuscript provides no explicit estimate quantifying the dependence of the constants on this separation distance. Without such a bound it remains unclear whether the inequalities remain uniform or controllable as the separation tends to zero; this dependence is load-bearing for the sharpness and applicability of the claimed functional inequalities on general hybrid scales.
minor comments (1)
  1. [Introduction] The introduction would benefit from a brief comparison table or diagram contrasting the new nonlocal seminorm with the classical Gagliardo seminorm and with the Riemann-Liouville derivative-based spaces on time scales.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The single major comment is addressed point by point below.

read point-by-point responses

  1. Referee: The positive separation of connected components is invoked to control mixed continuous-discrete interactions in the Gagliardo seminorm (see the statement of the Poincaré, embedding, and Hardy/CKN results). However, the manuscript provides no explicit estimate quantifying the dependence of the constants on this separation distance. Without such a bound it remains unclear whether the inequalities remain uniform or controllable as the separation tends to zero; this dependence is load-bearing for the sharpness and applicability of the claimed functional inequalities on general hybrid scales.

    Authors: We thank the referee for this observation. The positive-separation hypothesis is used precisely to ensure that the mixed continuous-discrete contributions to the Gagliardo seminorm remain controlled by a strictly positive lower bound on inter-component distances. In the proofs of the Poincaré-type inequality, the fractional Sobolev embedding, and the Hardy/CKN inequalities, the constants arise from integrals over the off-diagonal interaction domain and therefore depend implicitly on the minimal separation distance δ (typically through factors of the form C/δ^β with β depending on s and p). As δ tends to zero these constants deteriorate, which is consistent with the degeneration of the hybrid structure toward a purely continuous interval. We agree that an explicit statement of this dependence would improve clarity and applicability. In the revised version we will add a remark (and, if space permits, a short appendix) that tracks the constants through the existing estimates and states the precise dependence on δ, thereby addressing uniformity as δ → 0. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definitions and external comparisons yield independent functional inequalities.

full rationale

The paper introduces a Gagliardo-type seminorm via the product Lebesgue Δ-measure on the off-diagonal domain of arbitrary time scales, then derives Banach/reflexive/Hilbert properties and, under the stated geometric hypotheses (bounded hybrid scales with positive separation of components), proves Poincaré, Sobolev embedding, and Hardy/CKN inequalities. These steps rest on standard measure-theoretic arguments and explicit estimates rather than self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The obstruction to Riemann–Liouville equivalence on hybrid scales is shown by direct computation of mixed continuous–discrete cross terms, which is an independent verification rather than a tautology. The positive-separation hypothesis is a verifiable geometric assumption, not a hidden redefinition of the seminorm itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central contribution is a new seminorm definition relying on standard measure-theoretic properties of time scales; no free parameters are fitted, no new physical entities are postulated, and background axioms are drawn from existing time-scale calculus.

axioms (1)
  • standard math Existence and basic properties of the Lebesgue delta-measure on arbitrary time scales together with the induced product measure on the off-diagonal interaction domain.
    Directly invoked to define the nonlocal Gagliardo seminorm for fractional orders between 0 and 1.
invented entities (1)
  • Nonlocal Gagliardo seminorm on time scales no independent evidence
    purpose: To equip functions on arbitrary time scales with a genuinely nonlocal notion of fractional regularity.
    New object introduced by the paper; no independent falsifiable evidence outside the definition itself is supplied.

pith-pipeline@v0.9.0 · 5851 in / 1617 out tokens · 72168 ms · 2026-05-22T10:21:14.409840+00:00 · methodology

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