Lower bounds for the centered Hardy-Littlewood maximal operator on the real line

F.J. P\'erez L\'azaro

Lower bounds for the centered Hardy-Littlewood maximal operator on the real line

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Centered maximal operator on the real line has Lp norm strictly above 1

desk verdict A clean, honest proof of the Ivanisvili-Zbarsky conjecture for the centered Hardy-Littlewood maximal operator on R, with a new explicit constant; minor gaps only. read the letter →

arxiv 1908.08425 v2 pith:CLIR2SH3 submitted 2019-08-22 math.CA

F.J. Pérez Lázaro This is my paper · ORCID

classification math.CA MSC 42B25

keywords centeredHardy-LittlewoodmaximaloperatorlowerboundsLpnormreallinerisingsunlemmaiteratedone-sided

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 centered Hardy–Littlewood maximal operator $M$ replaces a function at each point by the largest average of $|f|$ over intervals centered at that point. The paper proves that on the real line, for every $10$ such that $\|Mf\|_{L^p(\mathbb{R})} \ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$ for all $f\in L^p(\mathbb{R})$. This answers a conjecture left open after earlier work established the bound only for $1

What carries the argument

The engine of the proof is the sequence $\gamma_n = g_n(0)$, where $g_n:[-1/2,\infty)\to[0,1]$ is defined recursively by $g_0=0$ and $g_n(t)= \frac{1+\int_0^{1+2t} g_{n-1}(u)\,du}{2(1+t)}$. The parameter $\gamma_n$ measures how much of the leftward average survives after $n$ iterations of the centered maximal operator: Lemma 2.7 states $M^n f \ge \gamma_n M_L f$ pointwise for every locally integrable $f$, where $M_L$ is the one-sided left maximal operator, the largest average of $|f|$ over intervals ending at the point and extending leftwards. Since $\gamma_n$ increases to $1$, iterating the centered operator eventually captures almost all of the one-sided maximal function. Lemma 2.9 then converts this into the distributional inequality $|\{M^n f>\lambda\}| \ge \frac{\gamma_n}{\lambda}\int_{\{f>\lambda\}} f$, which is stronger than the usual weak $(1,1)$ bound and is what forces the $L^p$ norm to grow.

What would settle it

Find one exponent $p\in(1,\infty)$ and one sequence $f_k\in L^p(\mathbb{R})$ with $\|Mf_k\|_p/\|f_k\|_p\to1$; the paper predicts the infimum of this ratio is at least $1+\varepsilon_p>1$. Alternatively, take any locally integrable $f\ge0$ and any $\lambda>0$ and compute both sides of the claimed distributional inequality $|\{M^n f>\lambda\}|\ge \frac{\gamma_n}{\lambda}\int_{\{f>\lambda\}} f$; a single counterexample to this inequality would disprove Lemma 2.9 and with it Theorem 1.1.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1: for every $1<p<\infty$ there is $\varepsilon_p>0$ such that $\|Mf\|_{L^p(\mathbb{R})} \ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$ for every $f\in L^p(\mathbb{R})$. The proof gives a quantitative value: writing $A_p$ for the best constant in the strong $(p,p)$ inequality for $M$, and $\gamma_n$ for a sequence that increases to $1$, one may take $(1+\varepsilon_p)^p = 1 + \left(\frac{A_p-1}{A_p^n-1}\right)^p \left[\left(\frac{\gamma_n p}{p-1}\right)^{1/p}-1\right]^p$ for large enough $n$. The key step is to compare the iterated maximal operator $M^n$ with the one-sided left maximal operator $M_L$: the inequality $M^n f \ge \gamma_n M_L f$ combined with the rising-sun lemma yields a distributional lower bound for $M^n f$ in terms of the distribution of $f$ itself, which then converts into the $L^p$ lower bound.

Load-bearing premise

The argument hinges on an exact formula, from the classical rising-sun lemma, for how often the one-sided leftward maximal operator exceeds a level; if that formula failed for ordinary locally integrable functions, the main lower bound would collapse.

Editorial extensions

If this is right

For every $p>1$, the $L^p$ operator norm of the centered Hardy–Littlewood maximal operator on $\mathbb{R}$ is strictly larger than $1$; the ratio $\|Mf\|_p/\|f\|_p$ cannot be made arbitrarily close to $1$ by any choice of $f$.
The lower bound is quantitative: choosing $n$ with $\gamma_n p/(p-1)>1$ yields the displayed expression for $\varepsilon_p$, so the gap is not merely existential.
For $1<p<2$, taking $n=1$ and $\gamma_1=1/2$ recovers the earlier bound $\|Mf\|_p \ge (p/(2(p-1)))^{1/p}\|f\|_p$, and the new argument extends it to all $p$.
Because nonconstant fixed points satisfying $Mf=f$ exist in dimensions $d\ge3$ for large $p$, the theorem shows the real line is a special case where expansion always wins.
The argument also applies to each iterate $M^n$: every iterate has a uniform $L^p$ lower bound with constant $(\gamma_n p/(p-1))^{1/p}>1$, a stronger statement than the single-operator bound.

Reading between the lines

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

The paper does not pursue it, but the comparison $M^n f\ge \gamma_n M_L f$ should transfer to any one-parameter family of averaging sets for which an exact distributional identity holds for the associated one-sided maximal function, yielding explicit lower constants.
One could also use Lemma 2.9 to derive lower bounds in rearrangement-invariant spaces: since it controls level sets of $M^n f$ by averages of $f$ over its own level sets, the argument is not intrinsically tied to the $L^p$ scale.
Because the formula for $\varepsilon_p$ involves the best constant $A_p$, plugging in any proven upper bound for $A_p$ would turn the existence result into a concrete numerical gap for each $p$.

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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

Verdict first: this is a sound, honest paper that settles the Ivanisvili-Zbarsky conjecture for the centered Hardy-Littlewood maximal operator on the real line. The headline is that there is a uniform epsilon_p > 0. The genuinely new part is the iteration inequality M^n f ≥ γ_n M_L f with explicit γ_n → 1, and the resulting explicit formula for epsilon_p. The existence part overlaps with Zbarsky's simultaneous and independent work, but the paper says so in plain terms and adds a quantitative bound, so the novelty claim is fair.

The proof is mostly self-contained. Lemma 2.7's induction is neat; the functions g_n are explicit and the Lagrange expansion argument that γ_n → 1 is believable, though it compresses a 'calculus exercise' that a referee would want spelled out. The delicate step is Lemma 2.9, where the rising sun identity for the one-sided maximal operator is used as an equality rather than just weak-type. The paper applies it to L^p functions, while the standard statement is for L^1. That is fixable by truncation, but the manuscript itself does not acknowledge the extension. I would call that a minor gap, not a fault in the argument.

The extraction of the norm bound in Theorem 1.1 via inequality (11) and the telescoping (12) is standard and correct. The constant depends on A_p, the best strong (p,p) constant, which is not known explicitly; this makes the result existential in practice, but still an honest quantitative statement relative to A_p.

Citation pattern looks appropriate. The dependence on Grafakos, Ivanisvili-Zbarsky, and Zbarsky is acknowledged. No circularity.

Who is this for? Specialists in maximal operators and anyone interested in sharp norm estimates of classical operators. It deserves a serious referee; with a request to expand the proof of Lemma 2.2(2) and to justify the rising sun identity for L^p, it should be accepted.

Referee Report

0 major / 5 minor

Summary. The paper proves that for every 1 < p < ∞ the centered Hardy-Littlewood maximal operator M on the real line satisfies the uniform lower bound ‖Mf‖_{L^p} ≥ (1+ε_p)‖f‖_{L^p} for all f ∈ L^p(ℝ), with an explicit (though not numerically computed) ε_p > 0. The proof introduces a sequence γ_n defined by a recursive integral formula, establishes the pointwise inequality M^n f ≥ γ_n M_L f, converts this via the exact rising-sun identity into a distributional lower bound, integrates by the layer-cake formula to obtain a lower bound for ‖M^n f‖_p, and finally transfers this to a lower bound for ‖M f‖_p using sublinearity and the known strong (p,p) bound of M. The manuscript is an extended and self-contained version of the methods of Ivanisvili and Zbarsky and affirmatively answers their conjecture for d = 1.

Significance. If correct, the result settles a conjecture of Ivanisvili and Zbarsky and strengthens the earlier partial result for 1 < p < 2 to all p. The main novelty is the precise pointwise inequality M^n f ≥ γ_n M_L f with monotone constants γ_n ↑ 1, which gives a quantitative iteration argument. The paper is notable for its clarity and for avoiding fitted parameters: the constants γ_n are generated by a recurrence, the constant ε_p is expressed explicitly through the Hardy-Littlewood bound A_p, and the proof of the key distributional estimate is fully exposed. The techniques are elementary and likely to be useful for related maximal operators.

minor comments (5)

[Section 1, Theorem 1.1] The phrase 'for every n ≥ 1 we can select' is too broad; the displayed expression for (1+ε_p)^p involves the term [(γ_n^p/(p−1))^{1/p} − 1]^p, which is not a real number when γ_n^p/(p−1) < 1 and p is not an integer. The statement should explicitly restrict to n sufficiently large so that γ_n^p/(p−1) > 1, as is correctly done in the proof.
[Section 2, Lemma 2.2] In the proof of part (2), the verification that the functions c_n satisfy the same recurrence as h_n is dismissed as 'a calculus exercise'. Since this explicit formula is the basis for the monotonicity γ_n ↑ 1, please include the short computation or provide a precise reference.
[Section 2, Lemma 2.9] The exact rising-sun identity |{M_L f > α}| = (1/α)∫_{M_L f > α} f is cited from Grafakos [2, p.93], but that reference typically states the result for integrable functions. The paper applies it to locally integrable functions in L^p; please add a sentence explaining the extension, for example by truncating f to f_N = f 1_{[-N,N]} and using monotone convergence.
[Section 2, proof of Lemma 2.7] The induction display contains two typographical errors: the term 'hF(x+h)' should read 'hF(x,h)', and the lower limit of the second integral should be x rather than 2y−x+h (the integral is over [x, 2y−x+h]). These do not affect the validity of the argument.
[Section 3, proof of Theorem 1.1] In the final line, 'γnp/(p−1)' should read 'γ_n^p/(p−1)'; the exponent p is missing. The condition for choosing n is γ_n^p/(p−1) > 1, not γ_n p/(p−1) > 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lower bound is derived from explicit constructions and standard external facts, with no fitted input renamed as a prediction.

full rationale

The paper's central claim, Theorem 1.1, is obtained by an explicit chain: Lemma 2.2 computes the auxiliary functions g_n by a closed formula and proves g_n(t) tends to 1 using Lagrange expansion, an external standard identity. The constants γ_n = g_n(0) are therefore computed, not fitted to the target inequality. Lemma 2.7 establishes the pointwise inequality M^n f ≥ γ_n M_L f by induction directly from the definition of the centered maximal operator and the recurrence defining g_n; no part of that lemma assumes the conclusion of Theorem 1.1. Lemma 2.9 uses the inclusion {M^n f > λ} ⊇ {f > λ} ∪ {M_L f > λ/γ_n}, which follows from Lemma 2.7 and M^n f ≥ f a.e., and then applies Riesz's rising sun lemma from Grafakos [2, p.93] in the form |{M_L f > α}| = (1/α)∫_{M_L f > α} f. That exact identity is an external, machine-checkable standard fact for the one-sided maximal operator and is not derived from the paper's own conclusion. Theorem 1.2 follows by multiplying the distributional inequality by pλ^{p-1} and integrating via the layer-cake formula, which is a standard integration argument. Theorem 1.1 then combines Theorem 1.2 with the triangle inequality and the known strong (p,p) bound of M, using the explicit constant A_p. The final expression for (1+ε_p)^p depends on A_p and γ_n, both of which are independently determined, and the positivity for large n follows from γ_n ↑ 1 as proved in Lemma 2.2 and Remark 2.5. There is no step where a parameter is fitted to a subset of data and then called a prediction, no load-bearing self-citation chain, and no uniqueness theorem imported from the author's prior work. The derivation is self-contained apart from standard external results, and no circularity is present.

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

No free parameters or invented entities appear. The central proof uses standard external facts: the exact rising sun lemma identity for the left maximal operator, the L^p boundedness of the centered maximal operator, and a Lagrange expansion identity from [1]. The sequence γ_n is defined in the paper and its limit is proved; it is not an ad hoc fitted quantity.

assumptions (3)

standard math Riesz rising sun lemma: for the one-sided left maximal operator, |{M_L f > η}| = (1/η)∫_{M_L f>η} f for nonnegative f.
Invoked in Lemma 2.9 to convert the level-set inclusion into the lower distributional bound. This exact identity is load-bearing; the usual weak-type inequality alone would not suffice.
standard math The centered maximal operator M is bounded on L^p(R) for 1<p<∞, with finite norm A_p>1.
Used in (12) to bound ||M^i f - M^{i-1}f||_p by A_p^{i-1}||Mf - f||_p, and in the definition of ε_p. This is a classical result.
standard math Lagrange expansion identity used to evaluate lim_n γ_n = 1.
Lemma 2.2 Part 3 relies on [1, p.206] to sum the series for γ_n; this underpins the strictness of the lower bound for sufficiently large n.

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Pith. "Pith review of Lower bounds for the centered Hardy-Littlewood maximal operator on the real line." pith.science (2026). https://pith.science/paper/CLIR2SH3

@misc{pith2026190808425,
  author       = {Pith},
  title        = {Pith review of: Lower bounds for the centered Hardy-Littlewood maximal operator on the real line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLIR2SH3}},
  note         = {Machine review of arXiv:1908.08425}
}

abstract

Let $1<p<\infty$. We prove that there exists an $\varepsilon_p>0$ such that for each $f\in L^p(\mathbb{R})$, the centered Hardy-Littlewood maximal operator $M$ on $\mathbb{R}$ satisfies the lower bound $\|Mf\|_{L^p(\mathbb{R})}\ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Ch. A. Charalambides, Enumerative Combinatorics, CRC Press, 2002

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Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004

L. Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004

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Ivanisvili, B

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Ivanisvili, S

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

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[9]

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2019 doi
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Stanley, Enumerative Combinatorics, Vol

R.P. Stanley, Enumerative Combinatorics, Vol. 2, ed.1. Cambridge Studies in Advanced Mathematics, 1999

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Pith tools

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

[1] [1]

Ch. A. Charalambides, Enumerative Combinatorics, CRC Press, 2002

work page 2002

[2] [2]

Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004

L. Grafakos, Classical and modern Fourier analysis, Prentice Hall, 2004

work page 2004

[3] [3]

Ivanisvili, B

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

work page 2017

[4] [4]

Ivanisvili, S

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

work page 2019

[5] [5]

Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect

S. Korry, Fixed points of the Hardy-Littlewood maximal operator, Collect. Math. 52 (2001) 289--294

work page 2001

[6] [6]

Lerner, Some remarks on the Fefferman-Stein inequality, J

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

work page 2010

[7] [7]

Mart\'in, J

J. Mart\'in, J. Soria, Characterization of rearrangement invariant spaces with fixed points for the Hardy-Littlewood maximal operator, Ann. Acad. Sci.Fenn. Math. 31 (2006) 39--46

work page 2006

[8] [8]

Melas, E.N

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

work page 2017