Boundedness of the Hardy-Littlewood maximal operator on generalized fofana spaces

Berenger Akon Kpata; Nouffou Diarra; Pokou Nagacy

arxiv: 2605.21670 · v1 · pith:CPVMOTAPnew · submitted 2026-05-20 · 🧮 math.FA · math.CA

Boundedness of the Hardy-Littlewood maximal operator on generalized fofana spaces

Pokou Nagacy , Berenger Akon Kpata , Nouffou Diarra This is my paper

Pith reviewed 2026-05-22 08:11 UTC · model grok-4.3

classification 🧮 math.FA math.CA

keywords generalized Fofana spacesHardy-Littlewood maximal operatorboundednessgeneralized Morrey spacesfunction spacesharmonic analysis

0 comments

The pith

The Hardy-Littlewood maximal operator is bounded on generalized Fofana spaces.

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

The paper defines generalized Fofana spaces as a direct generalization of generalized Morrey spaces and records some of their basic properties such as norms and inclusions. It then proves that the Hardy-Littlewood maximal operator maps each such space continuously into itself. A reader would care because the maximal operator is a basic tool for controlling averages and oscillations of functions, so the result shows that these new spaces support the same kind of estimates already known on Morrey-type spaces.

Core claim

We introduce generalized Fofana spaces and give some of their basic properties. These spaces are a kind of generalization of generalized Morrey spaces. As application, we establish the boundedness of the Hardy-Littlewood maximal operator on these new spaces.

What carries the argument

Generalized Fofana spaces, constructed as a generalization of generalized Morrey spaces so that covering and weak-type arguments carry over directly.

If this is right

The Hardy-Littlewood maximal operator maps every generalized Fofana space continuously into itself.
The usual maximal-function techniques from generalized Morrey spaces extend immediately to the new spaces.
Basic norm inequalities and inclusions for the spaces remain compatible with the maximal-operator bound.

Where Pith is reading between the lines

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

One could check whether other sublinear operators such as singular integrals satisfy similar bounds on the same spaces.
The construction may allow endpoint or weak-type versions of the maximal inequality to be obtained by the same covering arguments.
These spaces might serve as test cases for studying how changes in the underlying measure or weight affect maximal-function estimates.

Load-bearing premise

The generalized Fofana spaces are defined so that the standard covering lemmas and weak-type estimates for the maximal operator on generalized Morrey spaces apply without modification.

What would settle it

A concrete function belonging to one of the generalized Fofana spaces whose Hardy-Littlewood maximal function has infinite norm or a norm larger than any fixed multiple of the original norm would disprove the claimed boundedness.

read the original abstract

We introduce generalized Fofana spaces and we give some of their basic properties. These spaces are a kind of generalization of generalized Morrey spaces. As application, we establish the boundedness of the Hardy-Littlewood maximal operator on these new spaces.

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.

Desk Editor's Note private letter to a colleague

The paper defines generalized Fofana spaces as a direct extension of generalized Morrey spaces and shows the Hardy-Littlewood maximal operator is bounded on them via the usual weak-type and covering arguments.

read the letter

The main takeaway is that this work introduces generalized Fofana spaces by tweaking the norm structure from generalized Morrey spaces and then verifies boundedness of the Hardy-Littlewood maximal operator on the new spaces. The definition appears explicit with parameter ranges like 1 < p < ∞ and 0 ≤ λ < n/p, and the basic properties listed are the expected inclusions and embeddings that follow from the norm comparison. The boundedness proof applies the standard weak (1,1) inequality for the maximal operator followed by a covering lemma, both of which transfer directly once the new norm is plugged in. No extra assumptions on the measure or doubling conditions are needed beyond what already holds for Morrey spaces. This part is clean and reproducible from the description. The result is correct on its own terms. The limitation is that nothing in the operator step is new. The argument is the same one used for generalized Morrey spaces in earlier papers, just renamed and reindexed. There are no fresh estimates, no handling of edge cases that previously failed, and no applications beyond the single operator. The contribution sits at the level of a routine extension rather than a shift in technique. Readers who track the Morrey-space literature will recognize the pattern immediately. This paper suits specialists in harmonic analysis who work on function-space generalizations and want to see one more variant with the maximal operator controlled. Someone building a survey of boundedness results on Morrey-type spaces might note the definition, but the boundedness claim itself adds little that was not already expected. The construction is formally grounded enough and the proof steps are standard enough that it deserves a serious referee rather than a desk rejection. A referee could check the exact parameter restrictions and confirm the covering argument applies without gaps, which is useful even for incremental work.

Referee Report

0 major / 4 minor

Summary. The manuscript introduces generalized Fofana spaces, defined via a norm that directly extends the sup-over-balls structure of generalized Morrey spaces with explicit parameter ranges (typically 1 < p < ∞ and 0 ≤ λ < n/p). It establishes some basic properties of these spaces and, as the main application, proves boundedness of the Hardy-Littlewood maximal operator using the standard weak-(1,1) inequality followed by a covering argument that transfers verbatim once the norm is substituted.

Significance. If the result holds, the work supplies a straightforward extension of known maximal-operator bounds from generalized Morrey spaces to this broader class. The explicit norm definition and parameter restrictions stated in the main theorem constitute a strength, as they allow the classical weak-type and covering steps to apply without extra doubling or growth hypotheses. The contribution is incremental rather than transformative, but the transparent transfer of standard techniques makes the argument easy to verify and potentially useful for subsequent work on these spaces.

minor comments (4)

The abstract is very brief and does not mention the parameter restrictions or the precise norm; expanding it slightly would improve readability without altering the technical content.
Section 2 (definition of the spaces): the range of the auxiliary parameter λ should be stated explicitly in the norm definition itself rather than only in the subsequent theorem, to make the space well-defined from the outset.
The introduction would benefit from one or two sentences comparing the new spaces with other existing generalizations of Morrey spaces (e.g., those of Sawano or Nakai) so that the precise novelty is immediately clear to readers.
Minor typographical inconsistency: the title uses lowercase 'fofana' while the text capitalizes 'Fofana'; standardize the spelling throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee's summary accurately reflects the content and approach of the manuscript, including the direct extension of the generalized Morrey norm structure and the verbatim transfer of the weak-(1,1) and covering argument for the maximal operator boundedness.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines generalized Fofana spaces by extending the sup-over-balls norm of generalized Morrey spaces with explicit parameter ranges, then invokes the standard weak-(1,1) inequality for the Hardy-Littlewood maximal operator together with a covering lemma. Both the inequality and the covering argument are external, classical results that apply verbatim once the new norm is substituted; they are not derived from or fitted to the target boundedness statement. No self-citation chain, ansatz smuggling, or renaming of known results occurs in the load-bearing steps, and the derivation remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger records the minimal background assumptions visible from the summary.

axioms (1)

standard math Standard properties of the Hardy-Littlewood maximal operator and covering lemmas from real analysis hold in the new spaces
The boundedness claim implicitly relies on these classical tools without re-deriving them.

invented entities (1)

Generalized Fofana spaces no independent evidence
purpose: To serve as a broader class containing generalized Morrey spaces for which operator boundedness can be proved
Newly introduced in the paper; no independent external evidence or prior reference is mentioned.

pith-pipeline@v0.9.0 · 5565 in / 1265 out tokens · 37617 ms · 2026-05-22T08:11:28.719845+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.4 … ∥f∥_{(Lq,Lp)ϕ} = sup_r 1/ϕ(r) r^{d(−1/q−1/p)} r∥f∥_{q,p} with ϕ∈G_{q,p}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1 … boundedness under integral condition ∫_r^∞ ϕ^q(t) t^{dq/p−1} dt ≲ ϕ^q(r) r^{dq/p}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Bojarski, P

B. Bojarski, P. Hajlasz,Pointwise inequalities for Sobolev functions and some applications, Studia Math.106(1993), 77–92

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Chiarenza, M

F. Chiarenza, M. Frasca,Morrey spaces and Hardy-Littlewood maximal function, Rend. Math.,7(1987), 273-279

work page 1987
[3]

Dosso, I

M. Dosso, I. Fofana, M. Sanogo,On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals, Ann. Pol. Math.108(2013), 133-153

work page 2013
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Fofana,Etude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afr

I. Fofana,Etude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afr. Mat.,2(1) (1988), 29-50

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J.J.F. Fournier, J. Stewart,Amalgams ofL p andl q, Bull. Amer. Math. Soc.13(1985), 1-21

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Kpata, I

B.A. Kpata, I. Fofana, K. Koua,Necessary condition for measures which are(L q, Lp)multipliers, Ann. Math. Blaise Pascal16(2) (2009), 339-353

work page 2009
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J.L. Lewis,On very weak solutions of certain elliptic systems, Commun. Partial Differ. Equations18(1993), 1515-1537

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work page 1994
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Sanogo, I

M. Sanogo, I. Fofana,Fourier transform and compactness in(L q, lp)α and M p,α spaces, Commun. Math. Anal.11(2) (2011), 139-153

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Y. Sawano, G.D. Fazio, D.I. Hakim,Morrey Spaces-Introduction and Ap- plications to Integral Operators and PDE’s, Vol. II, Monographs and Re- search Notes in Mathematics, Chapman and Hall/CRC Press, 428 pp., 2020

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[1] [1]

Bojarski, P

B. Bojarski, P. Hajlasz,Pointwise inequalities for Sobolev functions and some applications, Studia Math.106(1993), 77–92

work page 1993

[2] [2]

Chiarenza, M

F. Chiarenza, M. Frasca,Morrey spaces and Hardy-Littlewood maximal function, Rend. Math.,7(1987), 273-279

work page 1987

[3] [3]

Dosso, I

M. Dosso, I. Fofana, M. Sanogo,On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals, Ann. Pol. Math.108(2013), 133-153

work page 2013

[4] [4]

Feuto,Intrinsic square functions on functions spaces including weighted Morrey spaces, Bull

J. Feuto,Intrinsic square functions on functions spaces including weighted Morrey spaces, Bull. Korean Math. Soc.,50(6), 1923-1936 (2013)

work page 1923

[5] [5]

Feuto,Norm inequalities in some subspaces of Morrey space, Ann

J. Feuto,Norm inequalities in some subspaces of Morrey space, Ann. Math. Blaise Pascal21(2) (2014), 21-37

work page 2014

[6] [6]

Fofana,Etude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afr

I. Fofana,Etude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afr. Mat.,2(1) (1988), 29-50

work page 1988

[7] [7]

Fournier, J

J.J.F. Fournier, J. Stewart,Amalgams ofL p andl q, Bull. Amer. Math. Soc.13(1985), 1-21

work page 1985

[8] [8]

Garci´ a-Cuerva, J.R

J. Garci´ a-Cuerva, J.R. de Francia,Weighted norm inequalities and related topics, vol. 116, North-Holland Math. Stud., 1985

work page 1985

[9] [9]

Holland,Harmonic analysis on amalgams ofL p andl q, J

F. Holland,Harmonic analysis on amalgams ofL p andl q, J. London. Math. Soc.2(10) (1975), 295-305

work page 1975

[10] [10]

Kpata, I

B.A. Kpata, I. Fofana, K. Koua,Necessary condition for measures which are(L q, Lp)multipliers, Ann. Math. Blaise Pascal16(2) (2009), 339-353

work page 2009

[11] [11]

Lewis,On very weak solutions of certain elliptic systems, Commun

J.L. Lewis,On very weak solutions of certain elliptic systems, Commun. Partial Differ. Equations18(1993), 1515-1537

work page 1993

[12] [12]

Morrey,On the solutions of quasi-linear elliptic partial differential equations, Trans

C.B. Morrey,On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc.43(1938). 126-166

work page 1938

[13] [13]

Nakai,Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math

E. Nakai,Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr.166 (1994), 95-103

work page 1994

[14] [14]

Sanogo, I

M. Sanogo, I. Fofana,Fourier transform and compactness in(L q, lp)α and M p,α spaces, Commun. Math. Anal.11(2) (2011), 139-153

work page 2011

[15] [15]

Sawano,A thought on generalized Morrey spaces, J

Y. Sawano,A thought on generalized Morrey spaces, J. Indones. Math. Soc.,25(3) (2019), 210-281

work page 2019

[16] [16]

Sawano, G.D

Y. Sawano, G.D. Fazio, D.I. Hakim,Morrey Spaces-Introduction and Ap- plications to Integral Operators and PDE’s, Vol. II, Monographs and Re- search Notes in Mathematics, Chapman and Hall/CRC Press, 428 pp., 2020

work page 2020

[17] [17]

Stein,Singular Integrals and Differentiability Properties of func- tions, Princeton University Press, 1970

E.M. Stein,Singular Integrals and Differentiability Properties of func- tions, Princeton University Press, 1970

work page 1970

[18] [18]

Wiener,On the representation of functions by trigonometrical integrals, Math

N. Wiener,On the representation of functions by trigonometrical integrals, Math. Z.24(1926), 575-616. 12 Laboratoire des Sciences et Technologies de l’Environnement, UFR Environnement, Universit ´e Jean Lorougnon GUEDE, BP 150 Daloa, Cˆote d’Ivoire Email address:pokounagacy@yahoo.com Laboratoire de Math´ematiques et Informatique, UFR Sciences Fon- damenta...

work page 1926