On Multi-linear Maximal Operators Along Homogeneous Curves

Ben Krause; Lars Becker

arxiv: 2508.09080 · v1 · submitted 2025-08-12 · 🧮 math.CA

On Multi-linear Maximal Operators Along Homogeneous Curves

Lars Becker , Ben Krause This is my paper

Pith reviewed 2026-05-18 23:20 UTC · model grok-4.3

classification 🧮 math.CA

keywords multi-linear maximal operatorshomogeneous polynomial curvessmoothing estimatesLebesgue space boundsharmonic analysismulti-linear inequalitiesaveraging operators

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The pith

The multi-linear maximal operator along a homogeneous polynomial curve with distinct degrees satisfies the expected L^p bounds whenever each p_j exceeds 1 and the sum of reciprocals is at most 1.

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

The paper proves that for a homogeneous polynomial curve with strictly increasing positive degrees, the supremum over positive r of the one-over-r integral from zero to r of the product of absolute values of functions each shifted by the corresponding coordinate of the curve is bounded in L^p norm by the product of the L to the p_j norms. This holds under the condition that the p_j are all greater than one and the sum of their reciprocals equals one over p which is at most one. A reader would care because these bounds control the size of multi-linear averages along curves and support further results on pointwise behavior of such operators. The argument relies on adapting a smoothing estimate to the specific polynomial form of the curve to obtain the necessary decay.

Core claim

For the homogeneous polynomial curve given by gamma(t) equals (a1 t to the d1, up to an t to the dn) with distinct positive integers d1 less than ... less than dn and nonzero coefficients ai, the inequality states that the L^p norm of the supremum over r greater than zero of one over r times the integral from zero to r of the product over i of absolute value of f_i at x minus gamma_i of t, dt, is at most C times the product of the L to the p_j norms of the f_j, where C depends only on the p_j and the curve.

What carries the argument

The smoothing estimate adapted to control oscillations and averages along the homogeneous polynomial curve with distinct degrees.

If this is right

The maximal operator is finite almost everywhere when the input functions belong to the stated Lebesgue spaces.
The result extends the boundedness from lower-order cases to the full multi-linear setting for these curves.
Pointwise convergence statements for the associated multi-linear averages along the curve follow as direct consequences.
The same smoothing approach yields uniform bounds independent of the particular nonzero coefficients a_i.

Where Pith is reading between the lines

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

The method could extend to curves that are perturbations of homogeneous polynomials if a comparable smoothing property holds.
Numerical verification on low-dimensional examples with explicit polynomial curves would give concrete evidence for the size of the constant C.
Connections to ergodic theory may arise by viewing the averages as multi-linear ergodic operators along polynomial flows.
The dependence of C on the degree vector d could be tracked explicitly to obtain dimension-free estimates in some regimes.

Load-bearing premise

The smoothing estimate applies directly to homogeneous polynomial curves with distinct positive degrees.

What would settle it

A choice of functions f_i in the respective L to the p_j spaces together with a specific homogeneous curve for which the left-hand side supremum integral produces an L^p function with infinite norm while the product of norms on the right is finite.

read the original abstract

Suppose that \[ \vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vec{\gamma}} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.

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

This paper extends single and bilinear maximal inequalities to the full multi-linear case for homogeneous polynomial curves with distinct degrees by adapting an existing smoothing estimate.

read the letter

The key point is that Becker and Krause establish the multi-linear maximal inequality along the curve gamma(t) = (a1 t^{d1}, ..., an t^{dn}) with 1 ≤ d1 < ... < dn. For p1,...,pn >1 and 1/p = sum 1/pj ≤1, they bound the L^p norm of the sup over r of the average integral of the product of |fi(x - gamma_i(t))| by the product of the individual L^{pj} norms. This is new; prior work handled fewer functions or different curve classes, and the abstract makes clear the result does not reduce to earlier theorems. They do this by adapting the smoothing estimate from Kosz-Mirek-Peluse-Wright rather than starting from scratch. That choice keeps the argument focused and reuses a tool that already controls certain oscillatory integrals. The adaptation appears to handle the distinct exponents by preserving the necessary derivative conditions across the phases. On the positive side, the statement is clean, the constant depends only on the p's and the curve coefficients, and there is no obvious circularity or self-referential fitting. The main soft spot is the transfer of the smoothing bound itself. The stress-test note flags that unequal degrees could affect Jacobian or Hessian lower bounds after rescaling in the multi-linear product setting. If the paper only cites the original estimate without spelling out the verification for all simultaneous non-vanishing conditions, a referee would want to see those details written out. Minor gaps like that are common in adaptations and do not sink the result if they can be filled. This is for readers already working on maximal operators, multi-linear harmonic analysis, or polynomial curves in one dimension. It is a modest but honest step forward rather than a broad reorganization of the field. The work shows clear engagement with the literature and a reproducible claim, so it deserves peer review even if revisions are needed on the adaptation steps.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that for a homogeneous polynomial curve γ(t) = (a1 t^{d1}, ..., an t^{dn}) with 1 ≤ d1 < ... < dn and ai ≠ 0, the multi-linear maximal operator satisfies ||sup_r (1/r) ∫_0^r ∏ |fi(x - γi(t))| dt ||_{L^p(R)} ≤ C ∏ ||fi||_{L^{pj}(R)} whenever pj > 1 and 1/p = ∑ 1/pj ≤ 1. The constant C depends on the pj and the curve. The proof reduces the claim to an adapted smoothing estimate taken from Kosz-Mirek-Peluse-Wright.

Significance. If the adaptation is fully justified, the result extends single-linear and bilinear maximal inequalities along curves to the multi-linear case with distinct degrees of homogeneity. This would be a useful addition to the literature on maximal operators in harmonic analysis, particularly for applications involving polynomial phases with varying scaling.

major comments (1)

[§3 (Smoothing estimate)] §3 (Smoothing estimate): The adaptation of the Kosz-Mirek-Peluse-Wright smoothing bound to the multi-linear product along γ with unequal di is invoked directly, but the manuscript does not supply an explicit verification that the required non-vanishing conditions on the relevant Jacobians or Hessians (after the multi-linear rescaling induced by the distinct exponents) continue to hold. Because the central L^p bound is obtained by transferring this estimate, the omission is load-bearing for the main theorem.

minor comments (2)

[Theorem 1.1] The dependence of C on the coefficients ai and the specific degrees di is stated but not quantified; a brief remark on how the constant scales with these parameters would improve clarity.
[Introduction] Notation for the vector-valued function γ and the multi-index p = (p1,...,pn) is introduced without a dedicated preliminary subsection; a short paragraph collecting all standing assumptions would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and will revise the paper to incorporate the requested explicit verification.

read point-by-point responses

Referee: §3 (Smoothing estimate): The adaptation of the Kosz-Mirek-Peluse-Wright smoothing bound to the multi-linear product along γ with unequal di is invoked directly, but the manuscript does not supply an explicit verification that the required non-vanishing conditions on the relevant Jacobians or Hessians (after the multi-linear rescaling induced by the distinct exponents) continue to hold. Because the central L^p bound is obtained by transferring this estimate, the omission is load-bearing for the main theorem.

Authors: We thank the referee for identifying this point. The non-vanishing conditions on the Jacobians and Hessians are preserved under the multi-linear rescaling because the curve has strictly increasing degrees of homogeneity 1 ≤ d1 < ⋯ < dn together with ai ≠ 0; the leading-order terms in the phase functions therefore remain non-degenerate after the diagonal rescaling induced by the distinct exponents. Nevertheless, we agree that an explicit verification strengthens the exposition. In the revised manuscript we will add a short lemma (or subsection in §3) that computes the relevant determinants after rescaling and confirms they are non-zero, thereby justifying the direct transfer of the Kosz-Mirek-Peluse-Wright smoothing estimate to the present multi-linear setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; relies on external smoothing estimate

full rationale

The paper derives the multi-linear maximal inequality by adapting a smoothing estimate from the independent external reference Kosz-Mirek-Peluse-Wright. The abstract explicitly identifies this as the main tool, with no self-citation, self-definition, or fitted-input reduction visible in the stated claim or derivation outline. The central bound does not reduce by construction to quantities defined inside the paper; the argument remains self-contained against the cited external benchmark. No load-bearing step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the adapted smoothing estimate and the geometric assumptions on the curve; no free parameters or new entities are introduced in the abstract statement.

axioms (1)

domain assumption Smoothing estimate adapted from Kosz-Mirek-Peluse-Wright applies to the given homogeneous polynomial curve
Invoked as the main tool to prove the maximal inequality.

pith-pipeline@v0.9.0 · 5746 in / 1251 out tokens · 27497 ms · 2026-05-18T23:20:56.351690+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

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

Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright [8]; ... Proposition 1.6 ... ∥∫∏fi(x−γi(2−kt))dt∥L1≤C2−cl∏∥fi∥Ln
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

homogeneous polynomial curve ... 1≤d1<...<dn

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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

17 extracted references · 17 canonical work pages · 1 internal anchor

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Bourgain

J. Bourgain. A nonlinear version of Roth’s theorem for sets of positive density in the real line. J. Analyse Math. 50 (1988), 169–181. pages 2

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Demeter, T

C. Demeter, T. Tao, C. Thiele. Maximal multilinear operators. Trans. Amer. Math. Soc. 360 (2008), no. 9, 4989–5042. pages 1

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[3]

Durcik, S

P. Durcik, S. Guo, J. Roos. A polynomial Roth theorem on the real line. Trans. Amer. Math. Soc. 371 (2019), no. 10, 6973–6993. pages 2

work page 2019
[4]

Durcik, J

P. Durcik, J. Roos. A new proof of an inequality of Bourgain. Math. Res. Lett. 31 (2024), no. 4, 1047–1060. pages 2

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non-flat

A. Gaitan, V. Lie. The boundedness of the (sub)bilinear maximal function along “non-flat” smooth curves. J. Fourier Anal. Appl. 26 (2020), no. 4, Paper No. 69, 33 pp. pages 1, 2 9

work page 2020
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G. Hong. Polynomial Szemer´ edi for sets with large Hausdorff dimension on the torus . arxiv, https://arxiv.org/pdf/2507.14407 pages 2

work page arXiv
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B. Hu, V. Lie. On the Curved Trilinear Hilbert Transform . https://arxiv.org/pdf/2308.10706 pages 2

work page arXiv
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D. Kosz, M. Mirek, S. Peluse, J. Wright. The multilinear circle method and a question of Bergel- son. arxiv, https://arxiv.org/abs/2411.09478 pages 3, 4

work page arXiv
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B. Krause. A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension . Preprint, https://arxiv.org/abs/1904.10562. pages 2

work page internal anchor Pith review Pith/arXiv arXiv 1904
[10]

B. Krause. On Polynomial Progressions Inside Sets of Large Dimension . arxiv, https://arxiv.org/abs/2508.04680 pages 2

work page arXiv
[11]

M. Lacey. The bilinear maximal functions map into Lp for 2/3 < p ≤ 1. Ann. of Math. (2) 151 (2000), no. 1, 35–57. pages 1

work page 2000
[12]

Lacey, C

M. Lacey, C. Thiele, Lp estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. of Math. (2) 146 (1997), no. 3, 693–724. pages 1

work page 1997
[13]

Lacey, C

M. Lacey, C. Thiele, On Calder´ on’s conjecture.Ann. of Math. (2) 149 (1999), no. 2, 475–496 pages 1

work page 1999
[14]

X. Li. Bilinear Hilbert transforms along curves, I. Analysis & PDE Vol. 6 (2013), No. 1, 197-220. pages 1

work page 2013
[15]

X. Li. L. Xiao. Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials. Amer. J. Math. 138 (2016), no. 4, 907–962. pages 1

work page 2016
[16]

non-flat

V. Lie. On the boundedness of the Bilinear Hilbert transform along “non-flat” smooth curves. American Journal of Mathematics, 137(2):313–363, 2015. pages 1

work page 2015
[17]

E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. pages 7 Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany Email address : becker@math.uni-bonn.de Departm...

work page 1993

[1] [1]

Bourgain

J. Bourgain. A nonlinear version of Roth’s theorem for sets of positive density in the real line. J. Analyse Math. 50 (1988), 169–181. pages 2

work page 1988

[2] [2]

Demeter, T

C. Demeter, T. Tao, C. Thiele. Maximal multilinear operators. Trans. Amer. Math. Soc. 360 (2008), no. 9, 4989–5042. pages 1

work page 2008

[3] [3]

Durcik, S

P. Durcik, S. Guo, J. Roos. A polynomial Roth theorem on the real line. Trans. Amer. Math. Soc. 371 (2019), no. 10, 6973–6993. pages 2

work page 2019

[4] [4]

Durcik, J

P. Durcik, J. Roos. A new proof of an inequality of Bourgain. Math. Res. Lett. 31 (2024), no. 4, 1047–1060. pages 2

work page 2024

[5] [5]

non-flat

A. Gaitan, V. Lie. The boundedness of the (sub)bilinear maximal function along “non-flat” smooth curves. J. Fourier Anal. Appl. 26 (2020), no. 4, Paper No. 69, 33 pp. pages 1, 2 9

work page 2020

[6] [6]

G. Hong. Polynomial Szemer´ edi for sets with large Hausdorff dimension on the torus . arxiv, https://arxiv.org/pdf/2507.14407 pages 2

work page arXiv

[7] [7]

B. Hu, V. Lie. On the Curved Trilinear Hilbert Transform . https://arxiv.org/pdf/2308.10706 pages 2

work page arXiv

[8] [8]

D. Kosz, M. Mirek, S. Peluse, J. Wright. The multilinear circle method and a question of Bergel- son. arxiv, https://arxiv.org/abs/2411.09478 pages 3, 4

work page arXiv

[9] [9]

B. Krause. A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension . Preprint, https://arxiv.org/abs/1904.10562. pages 2

work page internal anchor Pith review Pith/arXiv arXiv 1904

[10] [10]

B. Krause. On Polynomial Progressions Inside Sets of Large Dimension . arxiv, https://arxiv.org/abs/2508.04680 pages 2

work page arXiv

[11] [11]

M. Lacey. The bilinear maximal functions map into Lp for 2/3 < p ≤ 1. Ann. of Math. (2) 151 (2000), no. 1, 35–57. pages 1

work page 2000

[12] [12]

Lacey, C

M. Lacey, C. Thiele, Lp estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. of Math. (2) 146 (1997), no. 3, 693–724. pages 1

work page 1997

[13] [13]

Lacey, C

M. Lacey, C. Thiele, On Calder´ on’s conjecture.Ann. of Math. (2) 149 (1999), no. 2, 475–496 pages 1

work page 1999

[14] [14]

X. Li. Bilinear Hilbert transforms along curves, I. Analysis & PDE Vol. 6 (2013), No. 1, 197-220. pages 1

work page 2013

[15] [15]

X. Li. L. Xiao. Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials. Amer. J. Math. 138 (2016), no. 4, 907–962. pages 1

work page 2016

[16] [16]

non-flat

V. Lie. On the boundedness of the Bilinear Hilbert transform along “non-flat” smooth curves. American Journal of Mathematics, 137(2):313–363, 2015. pages 1

work page 2015

[17] [17]

E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. pages 7 Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany Email address : becker@math.uni-bonn.de Departm...

work page 1993