Power weight inequalities for spherical maximal functions

Andreas Seeger; Joris Roos; Marco Fraccaroli

REVIEW 2 major objections 4 minor 21 references

This paper claims a complete, up-to-endpoints description of the weighted L^p power-weight range for spherical maximal operators, for every set of admissible radii, expressed through the Legendre–Assouad dimension function of the radius set

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2026-08-02 22:13 UTC pith:2DJ5JNUZ

load-bearing objection This paper genuinely resolves the weighted spherical maximal problem for arbitrary dilation sets up to endpoints, with the Legendre–Assouad function as the governing object; the main theorem holds together, but several key kernel estimates are sketched rather than fully proved. the 2 major comments →

arxiv 2602.17613 v2 pith:2DJ5JNUZ submitted 2026-02-19 math.CA

Power weight inequalities for spherical maximal functions

Marco Fraccaroli , Joris Roos , Andreas Seeger This is my paper

classification math.CA MSC 42B2528A80

keywords power weight inequalitiesspherical maximal functionrestricted dilation setsLegendre–Assouad functionAssouad spectrumtype setweighted norm inequalitiesfractal dimensions

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims a complete solution, up to endpoint cases, to an open problem in weighted harmonic analysis: decide exactly for which exponents p and which power weights |x|^α the spherical maximal operator M_E is bounded on L^p(|x|^α), for every nonempty set E of admissible radii. The answer is governed by two dimension-like invariants of E: the upper Minkowski-type exponent β and the Legendre–Assouad function ν^♯_E. The permitted region in the (1/p, α/p) plane lies between the straight line U(p) = (d−1)(p−1)−β and the curve L(p) = (d−1)(p−2)−(ν^♯)^†((d−1)(p−1)), with p ≥ 1 + β/(d−1). The result matters because previous work settled only special classes of dilation sets, whereas here the full range of p and α is described by a single geometric function of E.

Core claim

Theorem 1.1 states that the closure of the type set of M_E on weighted L^p spaces is exactly the region {(1/p, α/p) : p ≥ p_β, L(p) ≤ α ≤ U(p)}, equivalently boundedness holds iff max{α+β, ν^♯((d−1)(p−2)−α)} ≤ (d−1)(p−1). The upper bound comes from a dyadic frequency decomposition of the spherical-average kernel and an entropy hypothesis; the lower bound comes from two families of test functions showing that M_E must be at least as large as the number of separated radii in small intervals dictates. Apart from endpoint cases in p and α, this settles the weighted boundedness question for arbitrary dilation sets.

What carries the argument

The central object is the Legendre–Assouad function ν^♯_E(ρ), a limsup of normalized entropy counts: for each scale δ, one maximizes |J|^{-ρ} N(E∩J,δ) over intervals J of logarithmic diameter between δ and 1, divides by log(δ^{-1}), and takes the limsup. It is convex, increasing, equals β for ρ ≤ 0 and equals ρ once ρ exceeds the quasi-Assouad dimension. Its generalized inverse (ν^♯)^† converts the covering geometry of E into the admissible range of α: the lower boundary of the type set is exactly where this inverse touches (d−1)(p−1). The paper also splits the spherical-average kernel into dyadic frequency pieces K^j_t whose Fourier decay rates carry the analytic part of the argument.

Load-bearing premise

The sufficiency half rests on the dyadic kernel estimates (5.9a,b)–(5.11a,b): the frequency pieces of the spherical-average kernel must decay at the stated rates 2^{-j(d-1)/2} and 2^{-j(d-3)/2}, which in turn require the bump functions to have enough vanishing moments and the surface measure to decay like (1+|ξ|)^{-(d-1)/2}; the p=1 and derivative cases of these estimates are only sketched as analogous.

What would settle it

Working directly in the proof, check the p=1 case of (5.9a,b) and the analogous derivative estimate (5.15b), which the paper says follow similarly. If either estimate carries an extra logarithmic factor in j, the j-sum in Proposition 5.1 would not close and the sufficiency direction would not hold as stated. A numerical or symbolic verification of these inequalities for d=2,3 would settle whether the argument is sound; alternatively, testing Lemma 4.1's power law on a set with strictly convex ν^♯ would test the sharpness of the lower boundary.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Every radius set now has a complete weighted boundedness description up to endpoints; the earlier open problem is resolved in full generality.
For the full set of all radii the formula recovers the classical range 1−d < α < (d−1)p−d, p > d/(d−1); for lacunary sets it recovers 1−d ≤ α < (d−1)(p−1), p > 1.
For Assouad-regular sets and finite unions of them, the type set is polygonal, with corners at scales determined by the quasi-Assouad dimensions of the components; for general sets it can be curved.
The entropy criterion is checkable: to decide whether a weighted inequality holds for a given E, it suffices to estimate covering numbers N(E∩J,δ) over logarithmic intervals, not to know the set's fine structure.
In the allowed range, M_E is bounded on the weighted space, so spherical means converge almost everywhere along E for functions in those weighted spaces.

Where Pith is reading between the lines

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

Because the proof uses only the Fourier-decay and moment structure of the spherical-average kernel, the same Legendre–Assouad criterion is likely to govern other oscillatory convolution maximal operators with comparable decay; the paper does not pursue this.
The theorem identifies the Legendre–Assouad function as the sharp geometric statistic for weighted problems. A natural next step is to construct explicit compact radius sets realizing every admissible convex function ν^♯, which would show all shapes permitted by the theorem actually occur.
The proof leaves endpoints such as α = 1−d and p = p_β open; sharpening the interpolation step in Lemma 5.3 and the j-summation in Proposition 5.1 is the obvious route to a full boundary description.
One could use the same two test-function families with optimized parameters to probe related endpoint questions, such as restricted weak-type estimates at the boundary of the type set.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

This paper genuinely resolves the weighted spherical maximal problem for arbitrary dilation sets up to endpoints, with the Legendre–Assouad function as the governing object; the main theorem holds together, but several key kernel estimates are sketched rather than fully proved.

read the letter

Quick take: this is the real thing. It closes the weighted spherical maximal problem for arbitrary dilation sets, up to the standard endpoint caveat, and pins down the Legendre–Assouad function as the object that governs the type set. The sharp lower boundary for non-Assouad-regular sets (e.g., {1+n^{-a}}, a≠1) is genuinely new; prior work only had necessary conditions in that range and a positive result above p_1.

What the paper does well: the necessary and sufficient conditions match exactly, the two-family Knapp constructions in §4 are clean and the exponent accounting checks out, and the reduction from global to local entropy via scaling (Lemma 2.1) is a neat way to get the global statement from compact sets. I verified the equivalence of the explicit and implicit type-set descriptions, the boundary values, and the cancellation of the k-dependence against hypothesis (5.2). Nothing in the proof looks fitted or circular.

The soft spots are real but not disqualifying. The sufficiency proof leans on frequency-localized kernel estimates (5.9a,b)–(5.11a,b) and the derivative bound (5.15b), and several of these are asserted with “analogous” or “follows by interpolation” rather than written out. The p=1 case of (5.9a) is sketched in a sentence. A referee should ask for full details there; if any decay rate were off by a power of 2^j, the converse half would collapse. But the stated rates are consistent with standard stationary-phase heuristics and the moment condition N_0 ≥ (d-1)/2 is the right one. The endpoint cases are excluded by design, and the result characterizes the closure of the type set — both honestly stated.

I could not find a load-bearing error. The paper is written for harmonic analysts working on maximal operators and fractal dimensions of dilation sets; for them it is a must-read. Send it to a serious referee, with instructions to scrutinize the sketched kernel estimates.

Referee Report

2 major / 4 minor

Summary. The paper studies the spherical maximal operator M_E with a general set of dilation radii E, acting on weighted L^p(|x|^alpha) spaces. The main result, Theorem 1.1, gives a complete description — up to endpoint cases — of the closure of the type set T_E = {(1/p, alpha/p) : M_E bounded on L^p(|x|^alpha)}. This description is in terms of the Legendre–Assouad function of E, with the region bounded by the two curves U(p) and L(p). The proof has two parts: in Section 4, two families of Knapp-type test functions are used to extract the necessary entropy condition, reducing it to the lower bound L(p) <= alpha; in Section 5, a frequency-localized decomposition of the spherical averaging kernel is used to prove sufficiency under the scale-invariant hypothesis (5.2). The paper also discusses Assouad regular sets and finite unions of such sets.

Significance. If correct, this settles, up to endpoint issues, an open problem of Duoandikoetxea and Seijo and gives the first complete weighted description for arbitrary dilation sets E. The result unifies the unweighted theorem of Seeger–Wainger–Wright with the earlier partial weighted results of Duoandikoetxea–Vega and Duoandikoetxea–Seijo. The lower-bound construction is particularly elegant: the two cases of Lemma 4.1 yield the exact entropy quantity appearing in the Legendre–Assouad function, and the exponent bookkeeping is internally consistent. The sufficiency proof, while relying on several standard frequency estimates, is organized so that the role of the entropy hypothesis (5.2) is transparent. The paper is a substantial contribution to the harmonic analysis of maximal operators with restricted dilations.

major comments (2)

[Section 5.1, Eq. (5.15b)] The estimate (5.15b) is load-bearing: it controls the derivative term in Lemma 5.4 and hence is needed for Proposition 5.1. The proof is omitted with the words 'completely analogous'. Since the statement is not a routine off-the-shelf estimate (it involves interpolation between p=1 and p=2 and an endpoint where the L^1 norm of dK^j_t/dt contributes), the authors should include the short argument or give a precise reference. I checked that the stated rate is consistent: for p=2 it follows from (5.11b) together with the 2^{-j} factor, and for p=1 from the shell-measure bound plus ||dK^j_t/dt||_1 = O(2^j), but the written proof should not leave this to the reader.
[Section 5, first paragraph] The reduction to the case alpha<0 and p <= 1+1/(d-1) is justified by saying that alpha >= 0 was already handled in [6]. The introduction, however, cites [6] as proving a positive result only for p > p_1. Since the region for alpha >= 0 and p in (p_beta, p_1] is nontrivial (the upper bound U(p) is positive there), the authors should state precisely which theorem in [6] covers this case, or provide the argument. If [6] does not cover it, the sufficiency proof would need to be extended.

minor comments (4)

[Lemma 5.3] The statement says 'Let alpha >= 1-d', but the proof interpolates between alpha = 1-d and alpha = 0 and therefore establishes the estimate only for 1-d <= alpha <= 0. Since Proposition 5.1 assumes alpha < 0, this does not affect the argument, but the statement should be corrected (or the proof extended).
[Section 4.2, after Eq. (4.7)] The displayed estimate uses 'sup_{s in E_R ∩ I}', but E_R has not been defined in the proof of Lemma 4.1; it should be 'sup_{s in E ∩ I}'.
[Section 3.2] The formula for L(p) for finite unions appears to be missing parentheses. As typeset, 'β_j γ_j −(d−1)(p−1)' is ambiguous and likely should read 'β_j(γ_j −(d−1)(p−1)) / (γ_j−β_j)', matching the formula from the Assouad regular case in Section 3.1.
[Section 5, first paragraph] The sentence 'We may also assume that alpha < 0 since the case alpha >= 0 was already handled in [6]' would benefit from a precise theorem number or page reference in the bibliography, especially because the introduction's summary of [6] is stated for p > p_1.

Circularity Check

0 steps flagged

No significant circularity: the necessary and sufficient directions are derived from independent test-function and Fourier estimates, not from the theorem's conclusion.

full rationale

The derivation chain in Theorem 1.1 is self-contained and non-circular. The necessary direction is proved by Lemma 4.1, which constructs explicit test functions f_Q and g_U and computes genuine lower bounds for the operator norm, yielding inequality (4.2) involving N(E∩I,2^{-j}) and |I|. Passing from (4.2) to the entropy condition (4.1) is a direct comparison with the definition of the Legendre–Assouad function ν♯, not an assumption of the desired conclusion. The sufficient direction is proved by Proposition 5.1, which takes the quantitative entropy hypothesis (5.2) — itself a direct consequence of the strict inequality (5.1) — and derives weighted L^p boundedness through frequency-localized kernel estimates (5.11a)–(5.11b), interpolation, and summation. No parameter is fitted and no 'prediction' is renamed from an input; the entropy hypothesis is an assumption about the geometry of E, not about the operator. The paper also recovers known benchmarks (unweighted case α=0, full maximal operator, lacunary sets, regular sets), confirming that the result is not vacuous or tautological. The self-citations to [2], [3], and [14] supply the Legendre–Assouad formalism and a reproduced calculation, but these are prior mathematical tools and not a load-bearing chain that reduces the theorem to itself. The briefly sketched endpoint cases (e.g., the p=1 case of (5.9a) and the 'analogous' proof of (5.15b)) are potential correctness risks, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The theorem introduces no fitted constants and no new objects. The only input is the dilation set E and the dimension functions defined from it (nu# from the authors' prior work). All substantive axioms are prior theorems or classical estimates. The result is a genuine two-sided characterization, not a parameter fit.

axioms (6)

standard math Classical decay of the spherical Fourier transform: |sigmahat(xi)| <~ (1+|xi|)^{-(d-1)/2}
Invoked in (5.11a) and the L^2 bounds in Lemma 5.4; classical estimate (e.g., Stein-Weiss [21]).
domain assumption Unweighted L^p boundedness of M_E for p > p_beta (and Bourgain's d=2 case)
Used in Section 5.1 for the f_low and f_high reductions; prior theorem of Seeger-Wainger-Wright [18] / Bourgain [4].
domain assumption Duoandikoetxea-Seijo Lemma 4 (Lemma 2.1 here): for alpha < 0, global L^p(w_alpha) bounds follow from uniform local bounds plus unweighted L^p boundedness
Cited from [6]; the engine of the global-to-local reduction in Section 5.
domain assumption Properties of the Legendre-Assouad function nu#: convex, increasing, nu#(rho) = beta for rho <= 0, nu#(rho) = rho for rho >= gamma, and the inverse identity nu#(rho) <= s iff rho <= (nu#)^†(s)
Recalled from the authors' own [2,3]; used in Section 2.2 and the boundary-value discussion. These are prior theorems, not derived here.
standard math Stein analytic interpolation and Stein-Weiss interpolation with change of measures
Used in Section 2.1 (convexity of T_E) and in Lemmas 5.3 and 5.4.
standard math Existence of a C_c^infinity bump phi_0 with vanishing moments of orders 1 through N_0, N_0 >> (d-1)/2, generating the frequency decomposition sigma_t = sum_j K^j_t
Standard Littlewood-Paley-type construction; the moment condition drives the decay estimate (5.11a).

pith-pipeline@v1.3.0-alltime-deepseek · 13715 in / 39345 out tokens · 342480 ms · 2026-08-02T22:13:08.888035+00:00 · methodology

0 comments

read the original abstract

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(|x|^\alpha)$ spaces. Aside from endpoint cases, a complete description of the allowable ranges of $p$, $\alpha$ is given in terms of the Legendre--Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.

Figures

Figures reproduced from arXiv: 2602.17613 by Andreas Seeger, Joris Roos, Marco Fraccaroli.

**Figure 1.** Figure 1: visualizes a typical set TE . Note that TE is not necessarily a polygon, since ν ♯ |[0,∞) can be any nonnegative increasing convex function such that ν ♯ (ρ) = ρ for ρ ≥ 1 (see [2, Thm. 1.2 (ii)]). α (d−1)p 1 p x1 xβ 1 1 −x1 −xβ α (d−1)p 1 p x1 xβ 1 1 −x1 −xβ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 2 linked inside Pith

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Pith/arXiv arXiv
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[1] [1]

Z., 297(3-4):1057–1074, 2021

Theresa Anderson, Kevin Hughes, Joris Roos, and Andreas Seeger.L p →L q bounds for spherical maximal operators.Math. Z., 297(3-4):1057–1074, 2021

2021

[2] [2]

A fractal local smoothing problem for the wave equation.Bull

David Beltran, Joris Roos, Alex Rutar, and Andreas Seeger. A fractal local smoothing problem for the wave equation.Bull. Lond. Math. Soc., 57(12):3667–3690, 2025

2025

[3] [3]

Spherical maximal operators with fractal sets of dilations on radial functions

David Beltran, Joris Roos, and Andreas Seeger. Spherical maximal operators with fractal sets of dilations on radial functions. arXiv:2412.09390. Studia Math., to ap- pear

Pith/arXiv arXiv

[4] [4]

Averages in the plane over convex curves and maximal operators.J

Jean Bourgain. Averages in the plane over convex curves and maximal operators.J. Analyse Math., 47:69–85, 1986

1986

[5] [5]

The spherical max- imal operator on radial functions.J

Javier Duoandikoetxea, Adela Moyua, and Osane Oruetxebarria. The spherical max- imal operator on radial functions.J. Math. Anal. Appl., 387(2):655–666, 2012

2012

[6] [6]

Weighted inequalities for some spherical maximal operators.Illinois J

Javier Duoandikoetxea and Edurne Seijo. Weighted inequalities for some spherical maximal operators.Illinois J. Math., 46(4):1299–1312, 2002

2002

[7] [7]

Spherical means and weighted inequalities.J

Javier Duoandikoetxea and Luis Vega. Spherical means and weighted inequalities.J. London Math. Soc. (2), 53(2):343–353, 1996

1996

[8] [8]

Fraser.Assouad dimension and fractal geometry, volume 222 ofCam- bridge Tracts in Mathematics

Jonathan M. Fraser.Assouad dimension and fractal geometry, volume 222 ofCam- bridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2021

2021

[9] [9]

Fraser, Kathryn E

Jonathan M. Fraser, Kathryn E. Hare, Kevin G. Hare, Sascha Troscheit, and Han Yu. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra. Ann. Acad. Sci. Fenn. Math., 44(1):379–387, 2019

2019

[10] [10]

Fraser and Han Yu

Jonathan M. Fraser and Han Yu. New dimension spectra: finer information on scaling and homogeneity.Adv. Math., 329:273–328, 2018

2018

[11] [11]

The critical weighted inequalities of the spherical maximal function.J

Juyoung Lee. The critical weighted inequalities of the spherical maximal function.J. Geom. Anal., 35(8):Paper No. 220, 17, 2025

2025

[12] [12]

Quasi-Assouad dimension of fractals.J

Fan L¨ u and Li-Feng Xi. Quasi-Assouad dimension of fractals.J. Fractal Geom., 3(2):187–215, 2016

2016

[13] [13]

Adam Nowak, Luz Roncal, and Tomasz Z. Szarek. Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions.Commun. Pure Appl. Anal., 22(7):2233–2277, 2023

2023

[14] [14]

Spherical maximal functions and fractal dimensions of dilation sets.Amer

Joris Roos and Andreas Seeger. Spherical maximal functions and fractal dimensions of dilation sets.Amer. J. Math., 145(4):1077–1110, 2023

2023

[15] [15]

Problems on spherical maximal functions

Joris Roos and Andreas Seeger. Problems on spherical maximal functions. Preprint, arXiv:2511.11283, 2025

Pith/arXiv arXiv 2025

[16] [16]

Attainable forms of Assouad spectra.Indiana Univ

Alex Rutar. Attainable forms of Assouad spectra.Indiana Univ. Math. J., 73(4):1331– 1356, 2024

2024

[17] [17]

Endpoint mapping properties of spherical maximal operators.J

Andreas Seeger, Terence Tao, and James Wright. Endpoint mapping properties of spherical maximal operators.J. Inst. Math. Jussieu, 2(1):109–144, 2003. POWER WEIGHT INEQUALITIES FOR SPHERICAL MAXIMAL FUNCTIONS 17

2003

[18] [18]

Pointwise convergence of spher- ical means.Math

Andreas Seeger, Stephen Wainger, and James Wright. Pointwise convergence of spher- ical means.Math. Proc. Cambridge Philos. Soc., 118(1):115–124, 1995

1995

[19] [19]

Elias M. Stein. Maximal functions. I. Spherical means.Proc. Nat. Acad. Sci. U.S.A., 73(7):2174–2175, 1976

1976

[20] [20]

Stein and Guido Weiss

Elias M. Stein and Guido Weiss. Interpolation of operators with change of measures. Trans. Amer. Math. Soc., 87:159–172, 1958

1958

[21] [21]

Stein and Guido Weiss.Introduction to Fourier analysis on Euclidean spaces

Elias M. Stein and Guido Weiss.Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. M.F., Department of Mathematics and Statistics, University of Massachusetts Lowell, Lowell, MA, USA Email address:marco fraccaroli@uml.edu J.R., Department of Mathematics and Statist...

1971