On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms

B{\l}a\.zej Wr\'obel; Maciej Kucharski; Mateusz Kwa\'snicki

arxiv: 2507.17510 · v2 · pith:WILEYHMFnew · submitted 2025-07-23 · 🧮 math.CA · math.FA

On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms

Maciej Kucharski , Mateusz Kwa\'snicki , B{\l}a\.zej Wr\'obel This is my paper

Pith reviewed 2026-05-22 12:55 UTC · model grok-4.3

classification 🧮 math.CA math.FA

keywords higher-order Riesz transformstruncated operatorsradial kernelsFourier transform boundsL2 contractivitysingular integralsdimension dependence

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The pith

The Fourier transform of the rescaling kernel b_{k,d} is bounded by 1 in modulus for every order k and dimension d.

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

The paper connects truncated higher-order Riesz transforms to the full versions through an explicit factorization into a convolution operator whose kernel is a rescaled radial function b_{k,d}. It establishes that this kernel is nonnegative only for orders one and two, while for orders three and higher its L1 norm grows without bound as dimension increases. The central result is that the Fourier transform of b_{k,d} satisfies a uniform bound of one in absolute value. This bound immediately produces an L2 contractivity statement comparing the truncated operator to the full operator when the input is radial, and the same comparison applies to a broader class of singular integrals with smooth kernels.

Core claim

By writing the truncated operator as R_k^t = M_k^t R_k, where M_k^t is convolution against the rescaled kernel b_{k,d}^t(x) = t^{-d} b_{k,d}(x/t), the authors reduce all norm questions to properties of the fixed radial function b_{k,d}. They prove b_{k,d} is nonnegative only when k equals 1 or 2, that the L1 norm of b_{k,d} tends to infinity with dimension when k is at least 3, and that |hat b_{k,d}(xi)| is at most 1 for every positive integer k. The last fact yields the contractive inequality ||R_k^t f||_2 <= ||R_k f||_2 for radial f, together with the corresponding statement for general smooth-kernel singular integrals.

What carries the argument

The factorization operator M_k^t realized as convolution with the explicit radial kernel b_{k,d} obtained by rescaling b_{k,d}^1; its Fourier transform bound transfers directly to operator-norm comparisons.

If this is right

The truncated operator satisfies a dimension-independent L2 contractive bound relative to the full Riesz transform whenever the input function is radial.
The same contractive comparison extends to any singular integral whose kernel is smooth enough to admit the same factorization.
For orders one and two the kernel b_{k,d} remains nonnegative and has L1 norm exactly one for every dimension.
For orders three and higher the kernel changes sign and its total variation grows unboundedly as dimension tends to infinity.

Where Pith is reading between the lines

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

The radial restriction may be removable by a more refined multiplier argument or by passing to spherical harmonics.
The explicit form of b_{k,d} makes it possible to test the Fourier bound numerically for moderate k and large d.
Analogous factorizations could be examined on other spaces such as the sphere or on manifolds where Riesz transforms are defined.

Load-bearing premise

The identity R_k^t equals M_k^t times R_k holds exactly, with M_k^t the convolution operator whose kernel is the rescaled radial function b_{k,d}^t.

What would settle it

A direct computation or numerical check showing |hat b_{k,d}(xi)| greater than 1 for some k, d, and frequency xi would falsify the L2 contractivity claim.

read the original abstract

Fix a positive integer $k$. Let $R_k$ be a higher order Riesz transform of order $k$ on $\mathbb{R}^d$ and let $R_k^t,$ $t>0,$ be the corresponding truncated Riesz transform. We study the relation between $\|R_k f\|_{L^p(\mathbb{R}^d)}$ and $\|R_k^t f\|_{L^p(\mathbb{R}^d)}$ for $p=1$, $p=\infty,$ and $p=2.$ We do this by analyzing the factorization operator $M_k^t$ defined by the relation $R_k^t=M_k^t R_k.$ The operator $M_k^t$ is a convolution operator associated with an $L^1$ radial kernel $b_{k,d}^t(x)=t^{-d}b_{k,d}(x/t),$ where $b_{k,d}(x):=b_{k,d}^1(x).$ We prove that $b_{k,d} \ge 0$ only for $k=1,2.$ We also show that for fixed $k\ge 3$, \[ \lim_{d\to \infty}\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=\infty. \] This contrasts with the cases $k=1,2$, where it is known that $\|b_{k,d}\|_{L^1(\mathbb{R}^d)}=1$. Finally, we show that for any positive integer $k$, the Fourier transform of $b_{k,d}$ is bounded in absolute value by $1.$ This implies the contractive estimate \[ \|R_k^t f\|_{L^2(\mathbb{R}^d)}\le \|R_k f\|_{L^2(\mathbb{R}^d)} \] and an analogous estimate for general singular integrals with smooth kernels for radial input functions $f.$

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 separates low-order and high-order behavior for these kernels with some new statements on sign and L1 growth, but the abstract narrows the L2 claim more than the multiplier bound requires.

read the letter

Two things stand out. The paper proves that the radial kernel b is nonnegative only for k=1 and 2, and that its L1 norm diverges to infinity with dimension when k is at least 3. It also shows that the Fourier transform of b satisfies |hat b| <=1 for every positive integer k. The divergence and the uniform Fourier bound are new relative to the cited work on Riesz transforms; the positivity for low orders recovers known facts and serves mainly as contrast. The factorization R_k^t = M_k^t R_k, where M is convolution against the rescaled b, is a direct way to move norm questions onto properties of this single function, and that reduction is carried out cleanly. The L2 consequence has a small phrasing issue. The abstract says the contractive estimate holds for radial input functions, yet a radial multiplier with absolute value at most 1 controls the L2 norm of the convolution operator for every function. If the factorization holds in general, the bound ||R_k^t f||_2 <= ||R_k f||_2 should follow for all f, not just radial ones. This looks like an imprecise statement rather than a substantive gap, but it is worth fixing. The three main claims are logically independent and avoid circular definitions or fitted quantities. The abstract alone does not let me verify every derivation step, but the statements are explicit enough to be checked. This is a technical note for specialists in harmonic analysis who work on dimension-free bounds or truncated singular integrals. A reader already interested in higher-order Riesz transforms or Calderon-Zygmund kernels would pick up concrete facts about the auxiliary operator M. The work is precise enough to deserve a serious referee who can read the full proofs.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the connection between higher-order Riesz transforms R_k and their truncations R_k^t on Euclidean space by factoring the latter as M_k^t composed with the former, where M_k^t is the convolution operator with the rescaled radial kernel b_{k,d}^t. It establishes three main results: the kernel b_{k,d} is nonnegative solely when k = 1 or 2; for k ≥ 3 the L^1 norm of b_{k,d} diverges to infinity with the dimension d; and the Fourier transform of b_{k,d} satisfies |ˆb_{k,d}(ξ)| ≤ 1 for every positive integer k. The last fact is used to deduce an L^2 contractivity estimate between the truncated and untruncated operators, stated for radial functions, together with an analogous statement for general smooth-kernel singular integrals.

Significance. If the stated results hold, the work clarifies the dimension dependence of norm comparisons for truncated singular integrals of higher order. The Fourier-multiplier bound supplies a clean, dimension-free L^2 estimate, while the L1-divergence results for k ≥ 3 indicate that no such dimension-free control can be expected in L^1 or L^∞ for higher orders. The explicit radial kernel and its Fourier transform constitute concrete, verifiable objects that could be useful for further study of singular-integral truncations.

major comments (1)

[Abstract] Abstract, final sentence: the claim that |ˆb_{k,d}(ξ)| ≤ 1 implies the contractive estimate ||R_k^t f||_2 ≤ ||R_k f||_2 specifically for radial input functions f conflicts with the general L^2 multiplier theorem. Because b_{k,d} is radial, its Fourier transform furnishes a multiplier m(ξ) with |m(ξ)| ≤ 1, so the convolution operator M_k^t satisfies ||M_k^t g||_2 ≤ ||g||_2 for every g ∈ L^2. Substituting g = R_k f therefore yields the same inequality for arbitrary f. The manuscript should either remove the radial restriction or explain why the factorization R_k^t = M_k^t R_k holds only on the radial subspace.

minor comments (2)

[Abstract] The sentence introducing the factorization ('We do this by analyzing the factorization operator M_k^t defined by the relation R_k^t = M_k^t R_k') would be clearer if it specified the precise function space on which the identity holds (e.g., Schwartz functions or all L^p).
A short introductory paragraph separating the three logically independent claims (positivity, L^1 divergence, and Fourier bound) and indicating which are new would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the careful reading that identified an inconsistency in the abstract. We address the major comment below and will incorporate the necessary changes.

read point-by-point responses

Referee: [Abstract] Abstract, final sentence: the claim that |ˆb_{k,d}(ξ)| ≤ 1 implies the contractive estimate ||R_k^t f||_2 ≤ ||R_k f||_2 specifically for radial input functions f conflicts with the general L^2 multiplier theorem. Because b_{k,d} is radial, its Fourier transform furnishes a multiplier m(ξ) with |m(ξ)| ≤ 1, so the convolution operator M_k^t satisfies ||M_k^t g||_2 ≤ ||g||_2 for every g ∈ L^2. Substituting g = R_k f therefore yields the same inequality for arbitrary f. The manuscript should either remove the radial restriction or explain why the factorization R_k^t = M_k^t R_k holds only on the radial subspace.

Authors: We agree with the referee that the L^2 multiplier theorem applies without restriction to radial functions. Since b_{k,d} is radial, its Fourier transform defines a bounded multiplier with |m(ξ)| ≤ 1, so M_k^t is a contraction on all of L^2. The factorization R_k^t = M_k^t R_k holds for general f because both operators admit Fourier-multiplier representations that commute in the usual way on the Schwartz class (and extend by density). The radial restriction appearing in the abstract and in the statement for general smooth-kernel singular integrals was included for expository simplicity in an earlier draft but is not required. We will remove the phrase “for radial input functions f” from the abstract and revise the corresponding theorem statement in the body to reflect the general case. If the proof for general singular integrals requires additional radial assumptions for technical reasons unrelated to the multiplier bound, we will add a clarifying remark; otherwise the restriction will be dropped there as well. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit definitions and independent proofs

full rationale

The paper explicitly defines the factorization R_k^t = M_k^t R_k with radial kernel b_{k,d}^t obtained by rescaling, proves non-negativity and L1-norm behavior of b_{k,d} by direct analysis, and establishes the Fourier multiplier bound |hat b_{k,d}(xi)| <=1 as a proved fact. The L2 contractivity then follows from standard multiplier theory applied to this bound. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or unverified self-citation chain; the derivation remains self-contained against external analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of the Fourier transform on R^d and the definition of higher-order Riesz transforms as Calderon-Zygmund operators; no free parameters or invented entities are introduced.

axioms (1)

standard math The Fourier transform of a radial L1 function is well-defined and the multiplier bound transfers to the operator norm on L2.
Invoked to obtain the L2 contractivity from |hat b| <=1.

pith-pipeline@v0.9.0 · 5896 in / 1521 out tokens · 42355 ms · 2026-05-22T12:55:49.155070+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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

We show that for any positive integer k the Fourier transform of b_{k,d} is bounded in absolute value by 1. This implies the contractive estimate ||R_k^t f||_2 ≤ ||R_k f||_2 ... for radial input functions f.

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

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Grafakos, Classical Fourier Analysis , Graduate Texts in Mathematics vol

L. Grafakos, Classical Fourier Analysis , Graduate Texts in Mathematics vol. 249, third edition, Springer Science+Business Media New York 2014

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F. Klein, Über die Nullstellen der hypergeometrischen Reihe, Math. Annalen 37 (1890), no. 4, 573–590

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Kucharski, B

M. Kucharski, B. Wróbel, A dimension-free estimate on 𝐿2 for the maximal Riesz transform in terms of the Riesz transform, Math. Annalen 386 (2023), 1017—1039

work page 2023
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Kucharski, B

M. Kucharski, B. Wróbel, J. Zienkiewicz, Dimension-free 𝐿𝑝 estimates for higher order maximal Riesz trans- forms in terms of the Riesz transforms, Analysis and PDE (2025), to appear, arXiv:2305.09279

work page internal anchor Pith review arXiv 2025
[5]

J. Liu, P. Melentijević, J.-F. Zhu, 𝐿𝑝 norm of truncated Riesz transform and an improved dimension-free 𝐿𝑝 estimate for maximal Riesz transform, Math. Ann. 389 (2024), 3513–3534

work page 2024
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Lee Lorch, M. E. Muldoon, Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions. III. Can. J. Math. 22(6) (1970): 1238–1265. DOI:10.4153/CJM-1970-142-1 14 MACIEJ KUCHARSKI, MATEUSZ KWAŚNICKI, AND BŁAŻEJ WRÓBEL

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Mateu, J

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Mateu, J

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Mateu, J

J. Mateu, J. Verdera, 𝐿𝑝 and weak 𝐿1 estimates for the maximal Riesz transform and the maximal Beurling transform, Math. Res. Lett. (6) 13 (2006), 957–966

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E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Prince- ton, (1970)

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J. Verdera, The Maximal Singular Integral: Estimates in Terms of the Singular Integral , in M.A. Picardello (ed.), Trends in Harmonic Analysis, Springer INdAM Series 3, 2013. Maciej Kucharski, Institute of Mathematics, University of Wrocław, Plac Grunwaldzki 2, 50-384 Wrocław, Poland Email address: maciej.kucharski@math.uni.wroc.pl Mateusz Kwaśnicki, Depa...

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[1] [1]

Grafakos, Classical Fourier Analysis , Graduate Texts in Mathematics vol

L. Grafakos, Classical Fourier Analysis , Graduate Texts in Mathematics vol. 249, third edition, Springer Science+Business Media New York 2014

work page 2014

[2] [2]

Klein, Über die Nullstellen der hypergeometrischen Reihe, Math

F. Klein, Über die Nullstellen der hypergeometrischen Reihe, Math. Annalen 37 (1890), no. 4, 573–590

work page

[3] [3]

Kucharski, B

M. Kucharski, B. Wróbel, A dimension-free estimate on 𝐿2 for the maximal Riesz transform in terms of the Riesz transform, Math. Annalen 386 (2023), 1017—1039

work page 2023

[4] [4]

Kucharski, B

M. Kucharski, B. Wróbel, J. Zienkiewicz, Dimension-free 𝐿𝑝 estimates for higher order maximal Riesz trans- forms in terms of the Riesz transforms, Analysis and PDE (2025), to appear, arXiv:2305.09279

work page internal anchor Pith review arXiv 2025

[5] [5]

J. Liu, P. Melentijević, J.-F. Zhu, 𝐿𝑝 norm of truncated Riesz transform and an improved dimension-free 𝐿𝑝 estimate for maximal Riesz transform, Math. Ann. 389 (2024), 3513–3534

work page 2024

[6] [6]

Lee Lorch, M. E. Muldoon, Peter Szego, Higher monotonicity properties of certain Sturm-Liouville functions. III. Can. J. Math. 22(6) (1970): 1238–1265. DOI:10.4153/CJM-1970-142-1 14 MACIEJ KUCHARSKI, MATEUSZ KWAŚNICKI, AND BŁAŻEJ WRÓBEL

work page doi:10.4153/cjm-1970-142-1 1970

[7] [7]

Mateu, J

J. Mateu, J. Orobitg, J. Verdera, Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels, Ann. Math. 174 (2011), 1429–1483

work page 2011

[8] [8]

Mateu, J

J. Mateu, J. Orobitg, C. Pérez, J. Verdera, New Estimates for the Maximal Singular Integral , Int. Math. Res. Not, (19) 2010 (2010), 3658–3722

work page 2010

[9] [9]

Mateu, J

J. Mateu, J. Verdera, 𝐿𝑝 and weak 𝐿1 estimates for the maximal Riesz transform and the maximal Beurling transform, Math. Res. Lett. (6) 13 (2006), 957–966

work page 2006

[10] [10]

https://dlmf.nist.gov/ , Release 1.2.4 of 2025- 03-15

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/ , Release 1.2.4 of 2025- 03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

work page 2025

[11] [11]

E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Prince- ton, (1970)

work page 1970

[12] [12]

E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7 (1982), pp. 359–376

work page 1982

[13] [13]

E. M. Stein, J. O. Strömberg, Behavior of maximal functions in ℝ𝑛 for large 𝑛, Ark. Mat. 21, (1983), 259–269

work page 1983

[14] [14]

Verdera, The Maximal Singular Integral: Estimates in Terms of the Singular Integral , in M.A

J. Verdera, The Maximal Singular Integral: Estimates in Terms of the Singular Integral , in M.A. Picardello (ed.), Trends in Harmonic Analysis, Springer INdAM Series 3, 2013. Maciej Kucharski, Institute of Mathematics, University of Wrocław, Plac Grunwaldzki 2, 50-384 Wrocław, Poland Email address: maciej.kucharski@math.uni.wroc.pl Mateusz Kwaśnicki, Depa...

work page 2013