On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient

Pedro Miguel Campos

REVIEW 4 major objections 6 minor 19 references

On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper builds fractional Sobolev-Orlicz spaces around the Riesz fractional gradient and proves they support a full PDE theory: embeddings, Poincaré inequality, compactness, and existence, uniqueness, and continuous dependence of…

desk verdict A genuinely new fractional Sobolev-Orlicz framework worth refereeing, but the proof chain rests on an unverified compatibility between the revised (A2) condition and the [HH19] results it invokes. read the letter →

arxiv 2412.06346 v1 pith:GIU4PWQQ submitted 2024-12-09 math.AP

Pedro Miguel Campos This is my paper · ORCID

classification math.AP MSC 46E3546E3035J6035R11

keywords fractionalSobolev-OrliczspacesRieszgradientgeneralizedOrliczSobolevembeddingPoincaréinequalitycompactquasilinearPDEcontinuousdependence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces $\Lambda^{s,A}_0(\Omega)$, obtained by completing test functions with respect to a norm that measures a function and its Riesz fractional gradient $D^s u$ in a generalized Orlicz space. The goal is to give nonlinear PDEs that involve $D^s$—natural models for nonlocal elasticity—the same functional-analytic toolbox that classical Sobolev spaces give local problems. The paper proves the fractional Sobolev embedding and Poincaré inequality, continuity of $D^s$ in the fractional parameter $s$, an interpolation inequality, and two compact embedding theorems. These tools yield existence and uniqueness of weak solutions to the quasilinear Dirichlet problem $-D^s\cdot(a(x,D^s u)D^s u)=F$, and continuous dependence of solutions as $s\to\sigma$. If the framework holds, it extends the Lions–Calderón spaces $\Lambda^{s,p}_0$ to problems with non-uniform, position-dependent growth.

What carries the argument

The carrying mechanism is the Riesz fractional gradient $D^s u=D(I^{1-s}u)$, which is, up to a constant, the unique rotationally and translationally invariant operator of order $s$ mapping scalars to vectors and generating the fractional Laplacian through $-D^s\cdot D^s u=(-\Delta)^s u$. The argument combines the fractional fundamental theorem $u=I_s(R\cdot D^s u)$ with boundedness of the Riesz potential, the Riesz transform $R$, and Calderón–Zygmund singular integrals on generalized Orlicz spaces $L^A$. The generalized $\Phi$-function $A(x,\ell)$ encodes position-dependent, non-power growth; the conditions $(\mathrm{Inc})_p$ and $(\mathrm{Dec})_q$ control its growth, while $(A0)$, $(A1)$, $(A2)$ make the harmonic-analysis estimates available.

What would settle it

Take a double-phase $\Phi$-function $A(x,\ell)=\ell^p+\alpha(x)\ell^q$ with $\alpha\in L^\infty$ but $\alpha\notin C^{d/(q-p)}$, and check numerically or analytically whether the compact embedding $\Lambda^{s,A}_0(\Omega)\hookrightarrow L^A(\Omega)$ still holds for some $s\in(0,1)$; a failure would show that $(A1)$ is load-bearing. Equivalently, look for a $\Phi$-function satisfying $(\mathrm{Inc})_p$ and $(\mathrm{Dec})_q$ but violating $(A2)$ on an unbounded domain for which the Poincaré inequality of Theorem 2 breaks down.

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Extended reading notes

Core claim

The central claim is that $\Lambda^{s,A}_0(\Omega)$ is a complete functional-analytic setting for PDEs built on the Riesz fractional gradient, in the same way that $W^{1,A}_0$ serves local problems. Under growth conditions $(\mathrm{Inc})_p$ and $(\mathrm{Dec})_q$ on the generalized $\Phi$-function $A$, together with three technical assumptions $(A0)$, $(A1)$ and $(A2)$, the paper establishes a Sobolev embedding into a companion Orlicz space, a Poincaré inequality with constant $C/(1-2^{-s})$, continuity of $D^s$ as $s\to\sigma$, a Gagliardo–Nirenberg-type interpolation inequality, and two compactness theorems. On this basis it proves that the quasilinear problem (23) has a unique weak solution for every $F\in\Lambda^{-s,A'}(\Omega)$, and that solutions depend continuously on $s$ as $s\to\sigma\in(0,1]$.

Load-bearing premise

The load-bearing premise is that the generalized $\Phi$-function $A$ satisfies the abstract technical conditions $(A1)$ and $(A2)$, which guarantee that $A$ and its inverse vary in a controlled pointwise way across the domain; the paper assumes these conditions rather than deriving them from simpler growth assumptions, and they are needed for the Riesz-potential and Calderón–Zygmund bounds that power the Sobolev, Poincaré, and compactness theorems.

Editorial extensions

If this is right

The Sobolev embedding $\Lambda^{s,A}_0(\Omega)\subset L^B(\Omega)$ controls lower-integrability Orlicz norms by the fractional gradient, exactly as in the classical case.
The Poincaré inequality makes $\|D^s u\|_{L^A}$ an equivalent norm on $\Lambda^{s,A}_0$, with a constant that stays uniform in $s$ apart from the factor $1/(1-2^{-s})$.
The compact embeddings give the precompactness needed for direct-method existence proofs and for extracting convergent subsequences when $s$ varies.
The interpolation inequality and continuity of $D^s$ in $s$ permit passing to limits $s_n\to\sigma$, which is what makes continuous dependence of PDE solutions possible.
The quasilinear problem (23) has a unique weak solution, and the solution map is continuous in the fractional parameter $s$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same machinery should extend to obstacle problems and variational inequalities in the Orlicz setting, as the author notes in Remark 6; the monotonicity and compactness tools already point in that direction.
The explicit $s$-dependence of the constants (such as $1/(1-2^{-s})$) suggests quantitative bounds on how fast solutions change as $s\to 0$ or $s\to 1$, which could be tested computationally.
Relaxing $(A1)$ and $(A2)$ would likely require a weighted generalized Orlicz theory, analogous to the role of Muckenhoupt weights in the variable-exponent case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, quick take on arXiv:2412.06346. The paper introduces Lambda^{s,A}_0(Omega), fractional generalized Sobolev-Orlicz spaces based on the Riesz fractional gradient. That is genuinely new and it puts together a useful package: Sobolev and Poincare inequalities, interpolation, continuity in s, compact embeddings, plus existence/uniqueness/continuous dependence for a quasilinear PDE. I think the reader's assessment is about right: conditional, not because of a discovered counterexample, but because the proof chain has an external dependency that is asserted rather than verified.

The main soft spot is the (A2) condition. Definition 2 adopts the revised (A2) from HHS23, while several of the cited harmonic-analysis results, notably [HH19, Cor 5.4.5] for the Riesz potential and [HH19, Thm 4.4.3] for dyadic operators, were proved under the old (A2). Remark 2 says the results 'hold true' for the new definition, but no proof or precise reference is given. That is load-bearing: Theorems 1, 2, 3, 5, and 7 all sit on those bounds. The paper's examples (variable exponent, double phase, log perturbation) all come from HH19/HHS23 and are likely fine, but the general compatibility claim needs to be settled. This is fixable, but it has to be fixed before the framework is fully dependable.

One thing I'd push back on: the reader flags Theorem 3's far-field constant as asserted without computation. I actually think the computation is there in the proof, via Lemma 2 bounding the L^A norm by the Lp/Lq norms of 1/|x|^{d+sigma} and then the explicit integral bound. It is a bit quick, but it is not missing. So that specific worry is lighter than it looks.

The PDE section is standard monotonicity plus the compactness machinery, and it seems fine. I don't see a hidden contradiction in the paper; it is coherent on its own terms. The heavy reliance on the author's prior work and the same small community is not by itself a problem, since the results are genuinely building on that work.

Who this is for: anyone working on nonlocal equations with nonstandard growth, especially in peridynamics-type models. It gives a ready-made functional-analytic toolbox. A serious referee should be engaged, mainly to force the (A2) compatibility issue into the open and to ask for the missing verifications. I would accept it for review, with an expectation of revision. Not a desk reject.

Referee Report

4 major / 6 minor

Summary. The paper introduces a new family of function spaces, Λ^{s,A}_0(Ω), defined as the completion of C_c^∞(Ω) with respect to the norm ‖u‖_{L^A(R^d)} + ‖D^s u‖_{L^A(R^d;R^d)}, where A is a generalized Φ-function satisfying growth conditions (Inc)_p and (Dec)_q and the technical hypotheses (A0), (A1), (A2). The main results are: a Sobolev embedding (Theorem 1), a fractional Poincaré inequality (Theorem 2), continuity of D^s in s (Theorem 3), an interpolation inequality (Proposition 2), compact embeddings (Theorems 4 and 5), and an application to existence, uniqueness and continuous dependence of solutions to a quasilinear Dirichlet problem driven by the Riesz fractional gradient (Theorems 6 and 7). The proofs rely on harmonic-analysis results for generalized Orlicz spaces from [HH19] and [HHS23].

Significance. If the main results are correct, the paper provides a useful functional-analytic framework that extends Lions–Calderón spaces Λ^{s,p}_0 to a generalized Orlicz setting with the Riesz fractional gradient. The structure is clear and the intended application to nonlocal problems with nonstandard growth is natural. Among the strengths: the use of the fractional fundamental theorem and the Riesz transform to reduce fractional-gradient estimates to classical Riesz-potential estimates; an explicit Poincaré constant depending on s in a transparent way; the construction of an interpolation inequality; and a complete variational existence and uniqueness argument for the PDE application. However, the proof chain is heavily dependent on an unverified compatibility between the revised condition (A2) taken from [HHS23] and the theorems from [HH19] that are invoked throughout. Several estimates, especially the far-field term in Theorem 3 and the interpolation step in Proposition 2, are asserted rather than proved in detail. These are load-bearing issues, not mere presentation problems.

major comments (4)

[Definition 2, Remark 2] The paper explicitly adopts the revised condition (A2) from [HHS23], while most harmonic-analysis results are quoted from [HH19], which were formulated with the original (A2). Remark 2 states that the [HH19] results used 'hold true' for the revised definition, but no proof or precise statement from [HHS23] is supplied. This is load-bearing: Theorem 1 invokes [HH19, Corollary 5.4.5], Theorem 2 invokes [HH19, Theorem 4.4.3], Theorem 4 invokes [HH19, Corollary 5.4.3], and Theorems 5 and 7 inherit these dependencies. Moreover, Examples 1–3 are justified by citations to [HH19], not by a check against the revised (A2). The authors must either prove that every [HH19] result used remains valid under the revised (A2), or replace each invocation by the corresponding theorem from [HHS23], and verify the examples under the revised condition.
[Theorem 3, equations (13)–(16)] The proof of the far-field bound in Theorem 3 is incomplete. The passage from ‖∫_Ω |u(y)|/|x−y|^{d+σ} dy‖_{L^A((Ω')^c)} to ∫_Ω |u(y)| ‖ |x−y|^{-(d+σ)} ‖_{L^A((Ω')^c)} dy is asserted with the phrase 'Jensen's inequality, Lemma 2, and [HH19, Lemma 3.7.7]', but no justification is given for moving the Luxemburg norm inside the integral or for replacing the L^A norm of the kernel by the maximum of its L^p and L^q norms. The displayed estimate max{(C/((d+σ)p−d))^{1/p}, (C/((d+σ)q−d))^{1/q}} ≤ max{(C/σ)^{1/p}, (C/σ)^{1/q}} ≤ max{1, C/σ} also omits the computation of the relevant norms. Since Theorem 3 is used in Theorem 5 and Theorem 7, this gap needs to be closed with a detailed derivation.
[Proposition 2, equation (21)] The proof of the interpolation inequality is not complete. The step ‖u + (−Δ)^{t/2}u‖_{L^A} ≤ C‖u‖_{L^A}^{(t−s)/t}‖(−Δ)^{t/2}u‖_{L^A}^{s/t} is justified only by 'an argument where we optimize the dilation of u', which is not carried out. In the following displayed inequality the exponent on ‖D^s u‖ appears as s/σ, which should presumably be σ/s for interpolation between 0 and s; no derivation is given. The relation D^σ u = R(−Δ)^{σ/2}u is also used without stating the relevant boundedness result for R. As (21) is a key tool for Propositions 3–4 and for Theorems 5 and 7, the proof must be supplied in full.
[Theorem 5 and proof] The statement of Theorem 5 allows σ to be an arbitrary limit point in [0,1], but the proof requires choosing 0 < s_* < 1 with s_* < s_n for every n; this is impossible when s_n → 0. If the theorem is intended only for σ > 0, that assumption must be stated, since the later application in Theorem 7 is for σ ∈ (0,1]. In the identification step, after taking limits one obtains ∫ v·φ = −∫ u div^σ φ, not 'div^{s_n} φ' as written; the displayed formula appears to contain a typo that obscures the argument.

minor comments (6)

[Theorem 5, statement] The conclusion states 'D^{s_n}u_n ⇀ Du in L^A(R^d)', but the limiting object should be D^σ u. Please correct this typo.
[Theorem 7, equation (24) and proof] Equation (24) has an unbalanced parenthesis in 'a(x,D^{s_n}u_n)D^{s_n}u_n · D^{s_n}w) dx'. In the limiting equation, 'D^σu_n' should read 'D^σu' and 'D^sσ w' should read 'D^σ w'.
[Proposition 4, proof] The notation 'L^A(R^d; R^s)' appears in the sentence 'D^{s_n}g_n → D^σu in L^A(R^d; R^s)'; the target space should be R^d. Also, the construction of the sequence {u_n} would benefit from a clearer definition of the indices q(k).
[Example 2] In the logarithmic perturbation example, the function A is written as ℓ^{p(x)} log(e+r), but the variable r is not defined; it should be ℓ. This makes the example hard to parse.
[References] In the reference [HHS23], the author name appears corrupted as 'Artur S/suppress labuszewski'; it should be corrected (presumably Artur Słabuszewski).
[Remark 3] The equality 'A(x,r)=Cr^p' when p=q requires also the normalization A(x,1) constant in x; as stated, the conclusion is immediate only with the additional condition (A0). This may be worth clarifying.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the results are deduced from independent harmonic-analysis theorems; the unproved compatibility of the revised (A2) condition is a correctness risk, not a circular reduction.

full rationale

The derivation chain is not circular. The spaces Λ^{s,A}_0(Ω) are defined as completions with respect to the norm ‖u‖_{L^A} + ‖D^s u‖_{L^A}, and the main embedding, compactness, interpolation, and PDE results are deduced from independent theorems in [HH19] and [HHS23] — boundedness of the Riesz potential, uniform L^A bounds for dyadic operators, and Calderón-Zygmund estimates — together with the representation formula (9) from [SS15]/[Pon16]. These are external, parameter-free results whose assumptions do not include the paper's target theorems. The appeal to the author's earlier [CR23] is explicitly as the result being generalized, not as justification for the new conclusions. The one genuinely load-bearing step that is not adequately supported is Remark 2: the paper replaces (A2) by the revised version from [HHS23] and asserts without proof that the [HH19] results 'hold true' for this definition. That is an unverified external dependency and a legitimate correctness concern, but it is not circularity: the quoted [HH19] theorems do not reduce to the paper's conclusions, and no fitted parameter or self-referential definition forces the results. Accordingly the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The paper introduces one new mathematical object, the space Lambda^{s,A}_0(Omega), plus its dual. The central claims rest on a chain of external harmonic-analysis results for generalized Orlicz spaces, on the technical conditions (A1) and (A2), and on the prior work of the author and collaborators. There are no numerically fitted parameters, but the growth exponents p and q are assumptions that shape all constants. The strongest dependency is on the unverified (A1)/(A2) conditions and on the boundedness results from [HH19].

free parameters (2)

The constants p and q in the (Inc)_p and (Dec)_q growth conditions = A parameter pair (p,q) with 1 < p <= q < infinity, e.g., p- and p+ in the variable-exponent example
Many of the theorems require A to satisfy (Inc)_p and (Dec)_q, and the results (especially Lemma 1 and the embeddings) depend on these exponents through the constants. They are not fitted to data but are chosen as hypotheses; for PDE applications with a(x,r) as in (5), they are determined by the bounds a- and a+.
The constant C in the inequalities = Generic, usually not explicit
The theorems state that various constants C are independent of s or u, but the proofs do not track the constants explicitly. The constant in Theorem 3 is written as depending on d, sigma, and the norm of chi_Omega in L^{A'}, so it is not uniform as sigma -> 0; this matches the statement but the strength of the result is sensitive to the value.

assumptions (6)

domain assumption The Riesz fractional gradient D^s defined on test functions by D^s phi = D(I^{1-s} phi), and the identification D^1 = D, D^0 = -R, are taken as the fundamental operators.
The whole paper builds on this operator and its properties, citing [SS15], [S20], [BCCS22].
domain assumption The generalized Phi-function A satisfies (A0), (A1), (A2) in addition to (Inc)_p and (Dec)_q.
These conditions are imposed throughout Theorems 1-5, 6, and 7. The paper does not prove them, it only cites [HH19] and [HHS23] for their validity in examples. They are explicitly stated as assumptions in Definition 2 and invoked, e.g., in the proofs of Theorems 1 and 2.
standard math The Riesz potential I^{1-s} and the vector Riesz transform R are bounded on the generalized Orlicz spaces L^A.
Used in the proofs of Theorems 1, 2, and Proposition 2, citing [HH19, Corollary 5.4.5 and 5.4.3]. These are external harmonic-analysis results that the paper treats as a black box.
standard math The dyadic averaging operators T_k are uniformly bounded on L^A.
Used in the proof of the Poincare inequality (Theorem 2), citing [HH19, Theorem 4.4.3].
domain assumption The space Lambda^{s,A}_0(Omega) can be defined as the completion of C_c^infty(Omega), and the Riesz fractional gradient extends by continuity.
Definition 4. This is a construction that requires the norm to be well-defined on test functions and the space to be complete; it is a sensible definition but it is an assumption that the resulting spaces have the stated structure, especially the duality in Proposition 1.
standard math The vector-valued Riesz transform satisfies R . R = -Id and the fractional fundamental theorem of calculus phi(x) = I_s(R . D^s phi)(x).
Used in the proofs of Theorem 1 and Proposition 2, citing [SS15] and [Gra14].

invented entities (2)

The fractional generalized Sobolev-Orlicz space Lambda^{s,A}_0(Omega)
purpose: The central object of the paper, a function space controlling both u and D^s u in a generalized Orlicz norm.
This is a new space defined by the paper. It does not correspond to an independently measurable physical entity; it is a mathematical construction whose properties are the subject of the paper. The space is a natural extension of known spaces, but its structure is not falsifiable outside the paper.
The dual space (Lambda^{s,A}_0(Omega))' = Lambda^{-s,A'}(Omega)
purpose: Used to formulate the PDE problem and the duality pairing in Proposition 1.
A formal construction; the paper proves a representation theorem for it, but it is a notational enlargement of the framework.

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Pith. "Pith review of On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient." pith.science (2026). https://pith.science/paper/GIU4PWQQ

@misc{pith2026241206346,
  author       = {Pith},
  title        = {Pith review of: On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIU4PWQQ}},
  note         = {Machine review of arXiv:2412.06346}
}

abstract

We introduce a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces $\Lambda^{s,A}_0(\Omega)$, where $A$ is a generalized $\Phi$-function satisfying the $(\mathrm{Inc})_{p}$ and $(\mathrm{Dec})_{q}$ conditions for $1<p\leq q<\infty$, as an extension of the Lions-Calder\'on spaces (also known as Bessel potential spaces) $\Lambda^{s,p}_0(\Omega)$ when $0<s<1$ to the generalized Orlicz framework. We obtain some continuous and compact embeddings for these spaces and study the continuous dependence of the Riesz fractional gradient $D^s$ with respect to $s\in[0,1]$ as $s\to \sigma\in[0,1]$. Finally, we apply these results to study the existence, uniqueness and continuous dependence of a family of partial differential equations depending on the Riesz fractional gradient as $s\to\sigma\in(0,1]$.

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Pseudodifferential and singular integral operators

Helmut Abels. Pseudodifferential and singular integral operators . De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012. An introduction with applications

work page 2012

[2] [2]

Comi, and Giorgio Stefani

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work page 2022

[3] [3]

Bellido, Javier Cueto, and Carlos Mora-Corral

Jos\' e C. Bellido, Javier Cueto, and Carlos Mora-Corral. Fractional P iola identity and polyconvexity in fractional spaces. Ann. Inst. H. Poincar\' e C Anal. Non Lin\' e aire , 37(4):955--981, 2020

work page 2020

[4] [4]

Bellido, Javier Cueto, and Carlos Mora-Corral

Jos\' e C. Bellido, Javier Cueto, and Carlos Mora-Corral. -convergence of polyconvex functionals involving s-fractional gradients to their local counterparts . Calculus of Variations and Partial Differential Equations , 60(1): Paper No. 7 , 2021

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Unilateral Problems for Quasilinear Operators with Fractional Riesz Gradients

Pedro M. Campos and Jos \'e F. Rodrigues. Unilateral problems for quasilinear operators with fractional R iesz gradients. arXiv preprint arXiv:2311.18428 , 2023

work page Pith review arXiv 2023

[6] [6]

Comi and Giorgio Stefani

Giovanni E. Comi and Giorgio Stefani. A distributional approach to fractional S obolev spaces and fractional variation: existence of blow-up. J. Funct. Anal. , 277(10):3373--3435, 2019

work page 2019

[7] [7]

Comi and Giorgio Stefani

Giovanni E. Comi and Giorgio Stefani. A distributional approach to fractional S obolev spaces and fractional variation: asymptotics I . Revista Matem \'a tica Complutense , pages 1--79, 2022

work page 2022

[8] [8]

Lebesgue and S obolev spaces with variable exponents , volume 2017 of Lecture Notes in Mathematics

Lars Diening, Petteri Harjulehto, Peter H\" a st\" o , and Michael R u z i c ka. Lebesgue and S obolev spaces with variable exponents , volume 2017 of Lecture Notes in Mathematics . Springer, Heidelberg, 2011

work page 2017