Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces

Ferit Gurbuz

REVIEW 4 major objections 4 minor 1 cited by

Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the Riesz potential $I_\beta$ is bounded from the homogeneous variable exponent Herz-Morrey-Hardy space $HM\dot{K}^{\alpha(\cdot),q_1}_{p_1(\cdot),\lambda}$ to the homogeneous variable exponent Herz-Morrey space…

desk verdict Routine template proof of a new Riesz potential bound, but the theorem as stated is internally inconsistent and the proof fails at allowed endpoints. read the letter →

arxiv 2411.13880 v1 pith:PKDRFWXE submitted 2024-11-21 math.FA math.AP

Ferit Gurbuz This is my paper · ORCID

classification math.FAmath.AP MSC 46E3542B2542B35

keywords RieszpotentialvariableexponentHerz-Morrey-HardyspacehomogeneousHerz-Morreyatomicdecompositionboundednesslog-Höldercontinuity

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 sets out to prove a norm inequality for the Riesz potential operator $I_\beta$ on spaces that combine variable-exponent Lebesgue behavior with Herz, Morrey, and Hardy structure. The main result asserts that $I_\beta$ maps the homogeneous variable exponent Herz-Morrey-Hardy space $HM\dot{K}^{\alpha(\cdot),q_1}_{p_1(\cdot),\lambda}$ into the homogeneous variable exponent Herz-Morrey space $M\dot{K}^{\alpha(\cdot),q_2}_{p_2(\cdot),\lambda}$, with the operator norm controlled by a constant times the input norm. The proof works by decomposing the input function into central atoms, applying dyadic estimates to each atom under the kernel of $I_\beta$, and summing the resulting geometric series using log-Hölder continuity of the exponent. If correct, the result extends earlier boundedness theorems for singular integrals on these spaces to fractional integrals.

What carries the argument

The load-bearing objects are the atomic decomposition of the homogeneous variable exponent Herz-Morrey-Hardy space (supplied by Theorem 1 of [5]) and the norm equivalence (2.8) that splits the Herz-Morrey norm into sums weighted by $\alpha(0)$ for nonpositive annuli and by $\alpha_\infty$ for positive ones. The dyadic estimates (3.2) and (3.3) control the action of $I_\beta$ on an atom supported in $B_j$ when measured on an annulus $F_k$, giving decay factors $2^{(\beta-n\delta_2)(k-j)}$ and $2^{(\beta-n\delta_1)(j-k)}$. These factors, together with the log-Hölder continuity assumptions, make the geometric series in the proof converge.

What would settle it

Test the borderline case $\alpha(\cdot)\equiv 2\lambda$: in the proof's $H_1$ estimate the inner sums become partial sums that grow linearly in $L$, so the supremum over $L$ is infinite, contradicting the claimed inequality unless a cancellation not present in the proof occurs.

Watch

Extended reading notes

Core claim

Theorem 2 states that for $0<q_1\le q_2<\infty$, $0<\lambda<\infty$, $0<\beta<n$, $0<p_1\le p_2<\infty$ with $p_1(\cdot),p_2(\cdot)\in \mathcal{B}(\mathbb{R}^n)$ and $\frac{1}{p_1(\cdot)}=\frac{1}{p_2(\cdot)}+\frac{\beta}{n}$, and for $\alpha(\cdot)\in L^\infty\cap \mathcal{P}^{\log}_0\cap \mathcal{P}^{\log}_\infty$ satisfying $2\lambda\le \alpha(\cdot)$ and $\beta-n\delta_2<\alpha(0)$, the Riesz potential $I_\beta$ is bounded from $HM\dot{K}^{\alpha(\cdot),q_1}_{p_1(\cdot),\lambda}$ to $M\dot{K}^{\alpha(\cdot),q_2}_{p_2(\cdot),\lambda}$. The proof represents a function $f$ as $\sum_j \lambda_j a_j$ with central $(\alpha(\cdot),p(\cdot))$-atoms supported in dyadic balls, bounds $\|(I_\beta a_j)\chi_k\|_{L^{p_2(\cdot)}}$ by geometric factors such as $2^{(\beta-n\delta_2)(k-j)}$ and $2^{(\beta-n\delta_1)(j-k)}$, and then sums over the atom index $j$ and the annulus index $k$ using the hypotheses on $\alpha(0)$ and $\alpha_\infty$.

Load-bearing premise

The proof repeatedly needs the strict inequality $\alpha_\infty > 2\lambda$ to make geometric sums converge, whereas the theorem states only $2\lambda \le \alpha(\cdot)$.

Editorial extensions

If this is right

If the theorem is correct, the Riesz potential gives a bounded embedding $HM\dot{K}^{\alpha(\cdot),q_1}_{p_1(\cdot),\lambda}\to M\dot{K}^{\alpha(\cdot),q_2}_{p_2(\cdot),\lambda}$ that mirrors the classical Sobolev-type index relation $1/p_2=1/p_1-\beta/n$.
The result extends the boundedness of singular integral operators on these spaces to the fractional integral case, filling the gap the introduction identifies.
The atomic-decomposition proof offers a route to boundedness for other convolution operators whose kernels obey a $|x-y|^{-n+\beta}$ size estimate, under the same index conditions.
The inequality is homogeneous in the function norm, so it yields a genuine operator norm bound rather than a conditional estimate.

Reading between the lines

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

I infer that the stated hypothesis $2\lambda\le\alpha(\cdot)$ is likely too weak: the displayed geometric sums require $\alpha_\infty>2\lambda$, so the theorem may need a strict inequality or a logarithmic factor in the borderline case.
The dyadic split through $\alpha(0)$ and $\alpha_\infty$ suggests that analogous boundedness should hold for fractional maximal operators or other Riesz-type kernels on the same spaces.
The two exponents $\delta_1,\delta_2$ from the log-Hölder estimates determine the admissible $\beta$ range, so improved estimates for characteristic functions would automatically widen the theorem's scope.

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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

Two things to know. This paper applies the standard atomic decomposition template to a genuinely new setting: Riesz potentials on homogeneous variable exponent Herz–Morrey–Hardy spaces. That gap is real, and the citations are appropriate. But the current version is not correct as stated, and the problems are not just cosmetic.

Theorem 2 contains a contradiction in its exponent hypotheses. It first assumes 1/p1(·) = 1/p2(·) + β/n, and then, as a condition for the known Riesz potential bound, requires 1/p2(·) = β/n − 1/p1(·). These two relations cannot both hold unless p1 = n/β. The second is almost certainly a typo for 1/p2(·) = 1/p1(·) − β/n, but as written the statement is ill-posed.

The proof has a more serious structural error. It applies the norm equivalence (2.8) to the q2-norm while raising to the q1 power. The equivalence is stated for the q-th power of the q-norm, so what is actually obtained is a bound for ||Iβ f||_{q2}^{q2}, not the q1-power that appears on the left of (3.4). The estimate therefore does not control the claimed target norm.

There is also a missing hypothesis on α∞. In the case 1 < q1 < ∞, the F1 and H1 estimates close with the condition α∞ > 2λ, written in parentheses at the end of those displays. Theorem 2 assumes only 2λ ≤ α(·), so α∞ = 2λ is allowed. At that endpoint the geometric series with terms 2^{(λ−α∞/2)jq1} has every term equal to 1 and diverges. The proof gives no alternative estimate at that endpoint. Similarly, the H2 estimate uses a condition β − nδ2 < α∞ that does not appear in the theorem's hypotheses.

The paper does show a sensible strategy and a clear attempt to fill a real gap. If the exponent relations were corrected, the norm equivalence applied with the right power, and the α∞ endpoint handled, a repaired version could be a small, useful note for specialists in variable exponent Herz–Morrey spaces. But as it stands, the central claim is not proved.

My recommendation is desk reject, with a detailed message to the author listing the three issues above. Do not send to referees in this form.

Referee Report

4 major / 4 minor

Summary. The paper claims a boundedness theorem for the Riesz potential I_β on homogeneous variable exponent Herz-Morrey-Hardy spaces. Specifically, Theorem 2 asserts that under exponent and log-Hölder conditions, I_β maps HM\dot{K}^{α(·),q_1}_{p_1(·),λ} into M\dot{K}^{α(·),q_2}_{p_2(·),λ}, with estimate (3.4). The proof uses the atomic decomposition of Xu and Yang, the norm equivalence (2.8), and a dyadic decomposition of the Herz-Morrey norm into terms F, G, H. The paper is essentially a proof of this single theorem, supplemented by definitions and recalled results.

Significance. If the theorem were correct, it would provide a modest extension of known Riesz-potential estimates to a combined variable-exponent Herz-Morrey-Hardy setting and would fill a gap noted in the introduction. The paper is also honest in relying on external results: the atomic decomposition is quoted from Xu and Yang [5], and the Lebesgue-space boundedness of I_β is quoted from Capone–Cruz-Uribe–Fiorenza [9]. However, the central theorem is not established as stated: the hypotheses are internally contradictory, the proof applies a norm equivalence with the wrong exponent, and several geometric-sum convergence steps require strict inequalities or use incorrect exponent arithmetic. Because these issues affect the main claim, the present version does not support its conclusion.

major comments (4)

[Theorem 2 (Section 3)] The hypotheses on the exponents are contradictory. The theorem first requires 1/p_1(·) = 1/p_2(·) + β/n, which is equivalent to 1/p_2(·) = 1/p_1(·) − β/n, the standard Sobolev relation for the Riesz potential. A few lines later it also requires the boundedness condition 1/p_2(·) = β/n − 1/p_1(·). These cannot hold simultaneously: adding the two equations gives 2/p_2(·) = 0. Thus Theorem 2, as written, has inconsistent hypotheses and is vacuous. The proof appears to use only the first relation, so the second clause is likely a typo, but the statement must be corrected before the claim can be evaluated.
[Section 3, proof of Theorem 2] The proof applies the norm equivalence (2.8) with the exponent q_1 to the norm of M\dot{K}^{α(·),q_2}_{p_2(·),λ}. However, (2.8) is stated for a space whose exponent equals the q appearing in the norm; replacing q_2 by q_1 gives the identity ‖I_β f‖_{M\dot{K}^{q_2}}^{q_1} ≈ max{ ... } that is not a consequence of (2.8). A correct argument would first use the monotonicity of ℓ^q norms (since q_1 ≤ q_2) to bound the q_2-norm by the q_1-norm, and only then apply (2.8) with q_1. The displayed equivalence near the beginning of the proof is therefore mathematically false as stated, although the argument might be repairable with an additional inequality.
[Section 3, F1 and H1 estimates] The estimates for F1 in the case 1 < q_1 < ∞ close with the parenthetical condition α_∞ > 2λ, and the same condition is used at the end of the H1 estimate. Theorem 2 assumes only 2λ ≤ α(·), so α_∞ = 2λ is explicitly allowed. At that endpoint the geometric series ∑_{j=0}^∞ 2^{(λ−α_∞/2)jq_1} has every term equal to 1 and diverges. Consequently the proof does not establish F1 ≲ Λ or H1 ≲ Λ for all f admitted by the theorem. The theorem would need either an added strict hypothesis α_∞ > 2λ or a different estimate at equality.
[Section 3, F2, G2 and H2 estimates] The dyadic estimates contain an exponent sign error. Substituting (3.2) into F2 and multiplying by the weight 2^{kq_1α(0)} gives a factor 2^{q_1(β−nδ_2+α(0))(k−j)}, not 2^{(β−nδ_2−α(0))(k−j)q_1} as written in the F2 estimate. The paper then uses the hypothesis β−nδ_2 < α(0) to make the displayed series converge, but the correct exponent would require the condition β−nδ_2+α(0) < 0, which is not assumed and is generally incompatible with β−nδ_2 < α(0). The same sign error recurs in the G2 and H2 estimates. These are load-bearing because they justify the boundedness of the 'far' dyadic contributions; without them the proof of (3.4) is incomplete.

minor comments (4)

[Throughout] The manuscript contains numerous typographical and grammatical errors, including 'V ARIABLE' and 'SP ACES' in the abstract; a thorough proofreading is needed.
[Theorem 2] The hypothesis '0 < p_1 ≤ p_2 < ∞' is unclear because p_1 and p_2 are variable exponent functions, not constants; the intended condition is presumably the pointwise inequality p_1(·) ≤ p_2(·).
[Equations (3.2), (3.3)] The notation α_j is used in the exponents of (3.2) and (3.3) but is never defined; the proof later splits terms into α(0) and α_∞, so the convention should be stated explicitly.
[Equations (3.1), (3.3)] The geometric condition '2|y| ≤ |x|' is claimed for j ≤ k−1, but for j = k−1 one only has |y| ≤ 2^j and |x| > 2^{j}, which does not imply 2|y| ≤ |x|; the dyadic separation should be stated with j ≤ k−2 or with a suitable constant.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the boundedness proof is a standard application of external atomic-decomposition and Lp boundedness theorems.

full rationale

The paper's central claim, Theorem 2, is proved by a direct atomic-decomposition argument. It invokes Xu and Yang's atomic characterization of homogeneous variable exponent Herz-Morrey-Hardy spaces (Theorem 1, cited as [5]) as an external black box, together with Izuki's estimates (2.4)-(2.6), variable-exponent Holder's inequality (2.7), and the known Lp-boundedness of the Riesz potential from Capone-Cruz-Uribe-Fiorenza [9]. None of these ingredients is the theorem being proved, and none is fitted or normalized so as to force the conclusion (3.4). The target norms are defined independently, and the proof reduces the desired inequality to estimates on atoms and geometric series whose convergence is controlled by the stated parameters. There are no self-referential definitions, no parameter fitted to the target data, and no load-bearing self-citation; references [7] and [8] by the author are not used in the proof. The proof does contain a hypothesis gap that is worth flagging: several displayed estimates, e.g. the F1 estimate for 1 < q1 < infinity and the H1 estimate, close with the parenthetical condition 'alpha_infty > 2lambda' although Theorem 2 only assumes 2lambda <= alpha(.). At alpha_infty = 2lambda the relevant geometric sums diverge, so the proof does not cover that endpoint. This is a correctness or completeness issue, not a circularity issue, because the missing condition is not hidden inside the definitions or the external theorems. Overall, no circular step can be identified from the paper's own equations or citations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No data fitting or invented entities appear. The paper relies on standard function-space theory and on cited external theorems. The main gaps are insufficient hypotheses and misapplied exponents, which are soundness issues rather than circularity.

assumptions (4)

domain assumption Atomic decomposition of HM K^{alpha(.),q}_{p(.),lambda}: every f in the space has a representation f = sum lambda_l a_l with central atoms (Theorem 1, from Xu and Yang [5]).
The proof relies on this decomposition without re-deriving it. It is cited from Xu and Yang (2015).
domain assumption Boundedness of the Riesz potential I_beta from L^{p1(.)} to L^{p2(.)} under 1/p2 = 1/p1 - beta/n (from Capone et al. [9]).
Invoked in (3.5) and in deriving (3.2). The paper's own statement of this condition contains a sign error; the standard condition is used in the proof.
domain assumption Estimates (2.4) and (2.5) for characteristic functions on balls in variable exponent Lebesgue spaces (from Izuki [10]).
These inequalities underlie the geometric factor estimates 2^{n delta (j-k)} used throughout the proof.
ad hoc to paper The proof assumes alpha_infty > 2 lambda (and at times alpha_infty > lambda) for convergence of geometric series, though Theorem 2 only states 2 lambda <= alpha(.).
Ends of the F1 and H1 estimates require strict inequality for convergence. This condition is not stated in the theorem and is not implied by the listed hypotheses.

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Cite this review

Pith. "Pith review of Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces." pith.science (2026). https://pith.science/paper/PKDRFWXE

@misc{pith2026241113880,
  author       = {Pith},
  title        = {Pith review of: Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PKDRFWXE}},
  note         = {Machine review of arXiv:2411.13880}
}

read the original abstract

In harmonic analysis, studies of inequalities of Riesz potential in various function spaces have a very important place. Variable exponent Morrey type spaces and the examines of the boundedness of such operators on these spaces have an important place in harmonic analysis and have become an interesting field. In this work, we obtain the boundedness of Riesz potential on homogeneous variable exponent Herz-Morrey-Hardy spaces under some conditions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The boundedness of rough generalized commutators with Lipschitz functions on homogeneous variable exponent Herz type spaces
math.AP 2025-06 conditional novelty 4.0 of 10

Generalized commutators with Lipschitz functions and rough kernels are shown to be bounded on homogeneous variable exponent Herz and Herz-Morrey spaces under log-Hölder conditions.

Reference graph

Works this paper leans on

11 extracted references · 9 canonical work pages · cited by 1 Pith paper

[5]

Xu, J., Yang, X.: Herz-Morrey-Hardy spaces with variabl e exponents and their applications. J. Funct. Spaces. 2015, Art. ID 160635, 1-19 (2015). doi: 10. 1155/2015/160635

work page 2015
[9]

Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractiona l maximal operator and fractional integrals on variable Lp spaces. Rev. Mat. Iberoam. 23, 743–770 (2007)

work page 2007
[1]

Czechoslovak Math

Kov´ aˇ cik, O., R´ akosn ´ ık, J.: On spacesLp(x) and W k,p(x). Czechoslovak Math. J. 41, 592-618 (1991). doi:10.21136/CMJ.1991.102493

work page arXiv 1991
[2]

M.: H p spaces of several variables

Feﬀerman, C., Stein, E. M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972) . doi:10.1007/BF02392215

doi:10.1007/bf02392215 1972
[3]

Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier trans- forms. J. Math. Mech. 18, 283–323 (1968/69)

work page 1968
[4]

Indiana Univ

Paluszy´ nski, M.: Characterization of the Besov spaces via the commutator operator of Coif- man, Rochberg and W eiss. Indiana Univ. Math. J. 44, 1–17 (199 5)

work page
[6]

Xu, J., Yang, X.: The molecular decomposition of Herz-Mo rrey-Hardy spaces with variable exponents and its application. J. Math. Inequal. 10, 977–10 08 (2016). doi:10.7153/jmi-10-79

work page doi:10.7153/jmi-10-79 2016
[7]

E˘ gitim Publishing (2 022)

G¨ urb¨ uz, F.: Some estimates for intrinsic square functions and commutators on homogeneous Herz spaces with variable exponents. E˘ gitim Publishing (2 022). (In Turkish)

work page

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[8]

Sultan, B.; Sultan, M., G¨ urb¨ uz, F.: BM O estimate for the higher order commutators of Marcinkiewicz integral operator on grand variable Herz-Mo rrey spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 72, 1000–1018 (2023). doi:10 .31801/cfsuasmas.1328691

2023
[10]

Izuki, M.: Boundedness of sublinear operators on Herz s paces with variable exponent and application to wavelet characterization. Anal. Math. 36, 3 3–50.(2010). doi: 10.1007/s10476- 010-0102-8

2010 doi
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M.: Harmonic Analysis

Stein, E. M.: Harmonic Analysis. Princeton University Press, Princeton, NJ, USA (1993). Department of Mathematics, Kırklareli University, Kırklar eli 39100, T ¨urkiye Email address : feritgurbuz@klu.edu.tr

1993

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Xu, J., Yang, X.: Herz-Morrey-Hardy spaces with variabl e exponents and their applications. J. Funct. Spaces. 2015, Art. ID 160635, 1-19 (2015). doi: 10. 1155/2015/160635

work page 2015

[2] [9]

Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractiona l maximal operator and fractional integrals on variable Lp spaces. Rev. Mat. Iberoam. 23, 743–770 (2007)

work page 2007

[3] [1]

Czechoslovak Math

Kov´ aˇ cik, O., R´ akosn ´ ık, J.: On spacesLp(x) and W k,p(x). Czechoslovak Math. J. 41, 592-618 (1991). doi:10.21136/CMJ.1991.102493

work page arXiv 1991

[4] [2]

M.: H p spaces of several variables

Feﬀerman, C., Stein, E. M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972) . doi:10.1007/BF02392215

doi:10.1007/bf02392215 1972

[5] [3]

Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier trans- forms. J. Math. Mech. 18, 283–323 (1968/69)

work page 1968

[6] [4]

Indiana Univ

Paluszy´ nski, M.: Characterization of the Besov spaces via the commutator operator of Coif- man, Rochberg and W eiss. Indiana Univ. Math. J. 44, 1–17 (199 5)

work page

[7] [6]

Xu, J., Yang, X.: The molecular decomposition of Herz-Mo rrey-Hardy spaces with variable exponents and its application. J. Math. Inequal. 10, 977–10 08 (2016). doi:10.7153/jmi-10-79

work page doi:10.7153/jmi-10-79 2016

[8] [7]

E˘ gitim Publishing (2 022)

G¨ urb¨ uz, F.: Some estimates for intrinsic square functions and commutators on homogeneous Herz spaces with variable exponents. E˘ gitim Publishing (2 022). (In Turkish)

work page