Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator

Samuel Zbarsky

REVIEW 2 major objections 4 minor 6 references

Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For $d=1$ or $d=2$ and every $p>1$, the centered Hardy--Littlewood maximal operator strictly raises the $L^p$ norm of every nonnegative nonzero function.

desk verdict A genuinely new fixed-point result for generic shapes and a nearly-true lower bound that is misstated for signed functions; fix the statement and it deserves refereeing. read the letter →

arxiv 1908.08487 v1 pith:6MVNHSG3 submitted 2019-08-22 math.CA

Samuel Zbarsky This is my paper · ORCID

classification math.CA MSC 42B2535J05

keywords centeredHardy-LittlewoodmaximaloperatorfixedpointsLplowerboundsconvexbodiessuperharmonicfunctionsGreen'sfunctionexpansionBesicovitchcoveringlemmafourthmoments

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

The paper proves two claims about the centered Hardy--Littlewood maximal operator $M$, defined by averaging over all scales and positions of a fixed centrally symmetric convex body $K$. First, when $d=1$ or $d=2$ and $10$ with $\|Mf\|_p \ge (1+\varepsilon)\|f\|_p$. Second, when $d\ge 3$ and $K$ satisfies an explicit nonzero fourth-moment condition (a generic condition), the infimum exponent $q_0(K)$ at which $M$ has an $L^p$ fixed point is strictly larger than the ball's value $q_0(B(0,1))=d/(d-2)$. The interest is that these results tie a strict lower bound on the operator norm to the absence of nonzero fixed points, and they show the ball is not the extremal shape for the onset of fixed points.

What carries the argument

The load-bearing object is the pointwise limit $f_*=\lim_{n\to\infty} M^n \mathbf{1}_{\delta_1 K}$ of iterates of the maximal operator applied to a small indicator shape; any nonzero $L^p$ fixed point of $M$ lies above a translate of this limit, so the limit's geometry controls every fixed point. In $d=1,2$ the proof mollifies a fixed point $g$ to a smooth fixed point $\tilde g$ and expands at small scales, $M\tilde g(x)\ge \tilde g(x)+\frac{\lambda^2}{2}\Delta \tilde g(x)+O(\lambda^3)$; the linear term disappears because $K$ is centered and normalized by $\int_K x_i x_j=\delta_{ij}$, so any point with $\Delta\tilde g(x)>0$ would violate the fixed-point inequality. Hence $\tilde g$ is superharmonic, and an existing classification argument shows a bounded superharmonic fixed point in $d=1,2$ must be constant. In $d\ge 3$ the same mollification shows fixed points are superharmonic and gives the lower bound $g\ge |x|^{2-d}$ on a large annulus; the refined step expands $Mh$ for $h(x)=|x|^{2-d}$ to fourth order, producing the term $\frac{\lambda^4}{24|K|}\sum_{i,j,k,\ell}\partial_{ijk\ell}h \int_K x_i x_j x_k x_\ell$. When that coefficient is nonzero, one maximal step pushes $Mh$ strictly above $h$ somewhere on the sphere $|x|=3$, and a harmonic comparison then improves the critical exponent beyond $d/(d-2)$.

What would settle it

Search in $\mathbb{R}^2$ for a centrally symmetric convex body $K$ and a bounded nonconstant $g$ with $Mg=g$: such a pair would contradict Theorem 3(1) and invalidate the proof of Theorem 1. A numerical check is to iterate $M^n\mathbf{1}_{\delta K}$ for a non-ellipsoidal $K$ in $\mathbb{R}^2$ and look for a nonconstant limiting profile, which would falsify the classification the theorem relies on.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of theorems. Theorem 1: in dimensions $d=1,2$, for every $1<p<\infty$ and every centrally symmetric convex body $K$, there is $\varepsilon=\varepsilon(K,p)>0$ such that $\|Mf\|_p\ge (1+\varepsilon)\|f\|_p$ for all nonnegative $f$. The proof also gives Theorem 3(1): in those dimensions the only $L^\infty$ fixed points of $M$ are constants, so in particular $M$ has no nonzero $L^p$ fixed points for any $1<p<\infty$. Theorem 3(2): for $d\ge 3$, if the fourth-moment integral $\int_K \sum_{i,j,k,\ell} a_{ijk\ell} x_i x_j x_k x_\ell$ is nonzero, where $a_{ijk\ell}$ are coefficients coming from fourth derivatives of the Green's function for Laplace's equation, then there is some $q>d/(d-2)$ such that $M$ has no fixed points in $L^p$ for $p\le q$. This condition holds for generic $K$ and in particular for the cube and the cross-polytope, so for generic shapes $q_0(K)>q_0(B(0,1))$.

Load-bearing premise

The load-bearing premise in dimensions $1$ and $2$ is that the only $L^\infty$ fixed points of the centered maximal operator are constant functions; if a nonconstant bounded fixed point existed for some convex shape, the iteration limit $\lim_{n\to\infty} M^n\mathbf{1}_{\delta_1 K}=1$ used throughout the lower-bound proof would fail.

Editorial extensions

If this is right

In $d=1,2$, the centered maximal operator has no nonzero $L^p$ fixed points for any $1<p<\infty$; in particular the equality $\|Mf\|_p=\|f\|_p$ can hold only for $f=0$, and every nonnegative nonzero function is strictly expanded.
For any fixed $K$ in $d=1,2$ and any fixed $p$, the proof gives a computable $\varepsilon(K,p)>0$; it does not, however, give asymptotics as $p$ varies, and the question of whether $\varepsilon$ can be chosen independent of $K$ is left open.
For $d\ge 3$, any shape satisfying the fourth-moment condition has an interval $(d/(d-2),q)$ free of $L^p$ fixed points, so $q_0(K)$ is strictly above the ball's critical exponent $d/(d-2)$.
The mechanism also shows that any shape with $q_0(K)=d/(d-2)$ would have to make all the higher-order moment conditions vanish, which singles out shapes whose even spherical harmonic coefficients are zero as the only potential exceptions to the generic conclusion.

Reading between the lines

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

Going beyond the paper, the same iteration argument suggests a testable numerical program: for a given $K$ in $\mathbb{R}^2$, measure the rate at which $M^n\mathbf{1}_{\delta K}$ approaches the constant function; extracting that rate could convert the existence of $\varepsilon(K,p)$ into quantitative asymptotics that the paper does not provide.
My inference is that the lower-bound property (2) is governed mainly by the absence of $L^\infty$ fixed points rather than by shape-specific details; if so, any construction of a counterexample in higher dimensions should focus on producing a nonconstant bounded fixed point.
In higher dimensions, the fourth-moment condition is only the first member of an infinite hierarchy described in the paper; a natural conjecture extending Theorem 3(2) is that $q_0(K)$ is determined by the first nonzero even-order coefficient in the Taylor expansion of the Green's function, with each nonzero coefficient pushing $q_0$ above the ball's value.

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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,

The genuinely new content here is the all-p lower bound for nonnegative functions in d=1,2 and the generic-shape fixed-point result in d≥3. The second theorem, Theorem 3(2), is the more interesting piece: it uses a quartic moment condition from the Green's function expansion to show that for most symmetric convex bodies the Lp fixed-point threshold sits strictly above d/(d-2). That is a real step beyond the ball case, and the argument from superharmonicity plus the boundary comparison on the annulus is coherent. The author also gives credit where the argument is borrowed: Korry's classification, the Ivanisvili–Jaye–Nazarov telescoping trick, and the earlier Zbarsky–Ivanisvili one-dimensional result are all cited.

The main theorem, however, is stated more broadly than it is proved. Definition (1) defines M without an absolute value, and Theorem 1 asserts the lower bound for all f in Lp. That is false for signed f: take f = -1_{B(0,1)}; every centered average is non-positive and tends to 0, so Mf ≡ 0 and the inequality fails for every p>1. The proof only works for nonnegative f. The mistake appears in the step after choosing the sets S_i, where the inequality Mf ≥ μ(1-δ1)/(1+δ1)^2 requires the integral of f over the difference region to be nonnegative, and later the step ∫(Mf)^p ≥ ∫f^p + ∫(Mf-f)^p uses Mf ≥ f pointwise. Both are fine when f ≥ 0 and false in general. The fix is easy: either define M with |f| in the usual way, in which case the theorem is plausible and the proof can be adapted in the obvious places, or restrict the theorem to nonnegative f. Either way the abstract and title need to say which.

A couple of smaller things. The convergence of the iterates M^n 1_{δK} to the constant 1 is asserted through Theorem 3(1), whose L∞ fixed-point classification is sketched rather than fully proved; the dependence of n on K and δ is not tracked. That is probably fixable, but the proof would be cleaner with a quantitative statement. The Besicovitch lemma in the appendix is a straightforward modification, fine. The quartic term calculation is terse; I'd want to see the Taylor expansion and the average over the sphere written out, but the idea is sound.

This paper deserves a serious referee. The fixed-point result for generic shapes is a genuine contribution, and the d=1,2 lower bound is valuable once it is stated for nonnegative or for |f|. I would not cite it in its current form; I would after the revision. Bring it to the reading group if you want a concrete example of a theorem statement running ahead of its proof.

Referee Report

2 major / 4 minor

Summary. The paper studies the centered Hardy-Littlewood maximal operator associated to a centrally symmetric convex body K, defined in Eq. (1) without an absolute value. The main result, Theorem 1, claims that for d=1,2 and 1<p<∞ there exists ε(K,p)>0 such that ||Mf||_p ≥ (1+ε)||f||_p for all f in Lp. Theorem 3 asserts that in d=1,2 the only L∞ fixed points of M are constant functions, while in d≥3 there are no Lp fixed points for p≤d/(d−2) and, for generic shapes, no fixed points for a range of p above that threshold, so that q0(K)>q0(B(0,1)). The proof of Theorem 1 combines a Besicovitch covering argument with iteration of the maximal operator and a contraction argument borrowed from [2]; the fixed-point results use mollification, Taylor expansion, and maximum-principle arguments. The paper is clearly written and the technical machinery is standard, but the statement of Theorem 1 is false for sign-changing functions as written.

Significance. If Theorem 1 is read as a statement about nonnegative functions, the paper gives a substantial improvement over the known range for the centered maximal operator: the lower bound is obtained for all p>1 in dimensions one and two, rather than only for p close to 1. The contraction argument and the Besicovitch covering step are clean and appear reproducible. The fixed-point genericity result for d≥3, if fully justified, is also new and interesting, and the quartic-moment criterion is explicit enough to be checked. However, the paper as written states Theorem 1 for arbitrary f, which is false; the proof works only for nonnegative f. This must be corrected before the result can be assessed as stated. The advertised generic-shape conclusion in Theorem 3(2) also needs a proof of genericity rather than an assertion.

major comments (2)

[§1, Eq. (1); §2, proof of Theorem 1] The definition of M in (1) lacks an absolute value and Theorem 1 is stated for arbitrary f∈Lp. This is false as stated: for f=−1_{B(0,1)}, every centered average is ≤0 and tends to 0 as λ→∞, so Mf≡0 and ||Mf||_p=0 for every 1<p<∞, while ||f||_p>0. The proof uses nonnegativity essentially: the step "Since K is convex ... Mf ≥ μ(1−δ1)/(1+δ1)^2 on x_i+δ1λ_iK" requires averages over the enlarged convex sets to be at least the average over S_i, and the final inequality ∫(Mf)^p ≥ ∫f^p+∫(Mf−f)^p requires Mf≥f pointwise. Please restrict Theorem 1 and the abstract to nonnegative f, or redefine M with |f| and rework the argument; the counterexample shows the present statement cannot stand.
[§3, Theorem 3(2) and following paragraph] The theorem asserts that for generic centrally symmetric convex bodies K in d≥3, q0(K)>q0(B(0,1)), but the proof establishes only the conditional statement: if the quartic moment ∫_K ∑ a_{ijkl} x_i x_j x_k x_l is nonzero, then no fixed points exist for some range p≤q with q>d/(d−2). The sentence "The condition in the statement above is generic" is not proved; the space of shapes, the topology, and the argument that the failure set is negligible (for example, contained in a proper algebraic hypersurface) should be given. In addition, the coefficients a_{ijkl} are described only by reference to derivatives of the Green's function, so an explicit formula or a precise reference would make the condition checkable. Without these additions, the advertised generic-shape conclusion is unsupported.

minor comments (4)

[§2, proof of Theorem 1] The passage "lim_{n→∞} M^n1_{δ1K} is a constant function (by Theorem 1)" should cite Theorem 3(1); as printed, it makes the proof of Theorem 1 appear circular.
[§2, fixed-point classification] The mollification argument shows Mg̃≤g̃, not equality; the sentence "so g̃ is also a fixed point of M" should be reworded to "so g̃ satisfies Mg̃≤g̃", since the subsequent expansion only needs the inequality.
[§2, Egorov step] The statement "by choosing n large enough" in the passage on M^{n−1}1_{δ1K} needs an explicit appeal to Egorov's theorem to justify uniform convergence off a small set on the compact set 2K.
[§3, after Eq. (6)] The phrase "Iterating (6)" is terse; a scaling argument for B(0,r) would clarify why the limit function is not in Lp.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs rely on external fixed-point, covering, and boundedness results and do not fit or rename the target inequality.

full rationale

The derivation chain in Theorem 1 is self-contained apart from genuine external inputs: the Besicovitch covering lemma (Lemma 5), L^p boundedness, and Korry's fixed-point classification [3], which the paper generalizes in Theorem 3(1) via mollification and a Taylor expansion. The line '(by Theorem 1)' in Section 2 is the only apparent self-reference; context shows this is a mislabeled citation to Theorem 3(1), which is explicitly introduced as the needed lemma and proved independently, so it does not make the argument circular. The later sentence 'We now copy the argument from [2]' borrows an elementary telescoping/subadditivity step, not a result that assumes the conclusion, and [2] proved only p<1.5, so this self-citation is not load-bearing. Section 3 derives the fourth-moment genericity condition from Taylor expansion of the Green's function rather than importing an ansatz. No parameter is fitted to the target inequality, and no known result is renamed as a new one. Hence score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. The proofs are self-contained except for standard external theorems and one covering lemma whose proof is only sketched.

assumptions (3)

domain assumption Besicovitch covering lemma for centrally symmetric convex bodies (Lemma 5)
The paper uses this to extract a bounded-overlap subcover in the level-set proof of Theorem 1. The proof is only sketched as a modification of the standard proof.
domain assumption Korry's classification of L∞ fixed points for the centered maximal operator (ball case)
Theorem 3(1) for general K is proved using Korry's argument from [3]; the paper does not reproduce that argument and relies on it.
standard math L^p boundedness of M (Hardy-Littlewood maximal theorem)
Used in §2 for the Lipschitz/telescoping argument and approximation by continuous functions.

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Pith. "Pith review of Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator." pith.science (2026). https://pith.science/paper/6MVNHSG3

@misc{pith2026190808487,
  author       = {Pith},
  title        = {Pith review of: Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MVNHSG3}},
  note         = {Machine review of arXiv:1908.08487}
}

abstract

For all $p>1$ and all centrally symmetric convex bodies $K\subset \mathbb{R}^d$ define $Mf$ as the centered maximal function associated to $K$. We show that when $d=1$ or $d=2$, we have $||Mf||_p\ge (1+\epsilon(p,K))||f||_p$. For $d\ge 3$, let $q_0(K)$ be the infimum value of $p$ for which $M$ has a fixed point. We show that for generic shapes $K$, we have $q_0(K)>q_0(B(0,1))$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[2]

Ivanisvili and S

P. Ivanisvili and S. Zbarsky. Centered Hardy–Littlewood maximal operator on the real line: Lower bounds. C. R. Math. Acad. Sci. Paris , 357(4):339–344, 2019

work page 2019
[1]

Ivanisvili, B

P. Ivanisvili, B. Jaye, and F. Nazarov. Lower bounds for uncentered maximal functions in any dimension. Int. Math. Res. Not. IMRN , (8):2464–2479, 2017. 7

work page 2017
[3]

S. Korry. Fixed points of the Hardy-Littlewood maximal operator. Collect. Math. , 52(3):289294, 2001

work page 2001
[4]

A. K. Lerner. Some remarks on the Feﬀerman-Stein inequality. J. Anal. Math., 112:329– 349, 2010

work page 2010
[5]

P. Mattila. Geometry of sets and measures in Euclidean spaces , volume 44 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectiﬁability

work page 1995
[6]

A. D. Melas and E. N. Nikolidakis. Local lower norm estimates for dyadic maximal operators and related Bellman functions. J. Geom. Anal., 27(3):1940–1950, 2017. 8

work page 1940

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Ivanisvili and S

P. Ivanisvili and S. Zbarsky. Centered Hardy–Littlewood maximal operator on the real line: Lower bounds. C. R. Math. Acad. Sci. Paris , 357(4):339–344, 2019

work page 2019

[2] [1]

Ivanisvili, B

P. Ivanisvili, B. Jaye, and F. Nazarov. Lower bounds for uncentered maximal functions in any dimension. Int. Math. Res. Not. IMRN , (8):2464–2479, 2017. 7

work page 2017

[3] [3]

S. Korry. Fixed points of the Hardy-Littlewood maximal operator. Collect. Math. , 52(3):289294, 2001

work page 2001

[4] [4]

A. K. Lerner. Some remarks on the Feﬀerman-Stein inequality. J. Anal. Math., 112:329– 349, 2010

work page 2010

[5] [5]

P. Mattila. Geometry of sets and measures in Euclidean spaces , volume 44 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectiﬁability

work page 1995

[6] [6]

A. D. Melas and E. N. Nikolidakis. Local lower norm estimates for dyadic maximal operators and related Bellman functions. J. Geom. Anal., 27(3):1940–1950, 2017. 8

work page 1940