Improved $L^p$ bounds for the strong spherical maximal operator

Jonathan Hickman; Joshua Zahl

Improved $L^p$ bounds for the strong spherical maximal operator

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Pith's one-line read This paper proves that the strong spherical maximal operator is bounded on $L^p(\mathbb{R}^n)$ for every $p>2$ in every dimension $n\geq 3$, matching the conjectured sharp threshold in dimension three.

desk verdict Genuine sharp-range advance for n=3 with a new incidence-geometric volume estimate; the flagged Pawlucki uniformity worry does not survive contact with the proof. read the letter →

arxiv 2502.02795 v1 pith:VQSIQNGU submitted 2025-02-05 math.CA

Jonathan Hickman , Joshua Zahl This is my paper · ORCID

classification math.CA MSC 42B2514P10

keywords strongsphericalmaximaloperatormultiparameteraveragesL^pestimatesellipsoidalannulidiscretisedincidencegeometryvariableslicingargumentsemi-algebraiccelldecompositionL2dualitymethod

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

This paper proves that the strong spherical maximal operator—the maximal average over concentric axis-parallel ellipsoids with arbitrary radii—is bounded on $L^p(\mathbb{R}^n)$ for every $p>2$ and every dimension $n\geq 3$. The range matches the conjectured sharp threshold $p>(n+1)/(n-1)$ in dimension three, where the threshold equals $2$. The proof works by isolating a discretised incidence-geometry estimate: after discarding a small set of exceptional higher-order tangencies between pairs of ellipsoids, the intersection of two thin ellipsoidal annuli has volume controlled by $\delta^2/(\delta+|t_1-t_2|)$. This estimate is then interpolated with earlier Fourier-analytic $L^p$–Sobolev bounds, improving the previously known range $p>2(n+1)/(n-1)$ and settling the $n=3$ case of the strong spherical maximal conjecture.

What carries the argument

The load-bearing object is the sublevel set $\Omega_k(t,r;\rho)$ of the polynomial map $\Phi_k(\omega,t,r)$ defined in (3.4), whose zero set encodes the higher-order tangency condition that the gradients $\nabla F_{0,1}(\omega)$ and $\nabla F_{x,r}(\omega)$ are parallel. The decisive fact is Lemma 3.4: $\Omega_k(t,r;\rho)$ can be covered by $O(1)$ sets of diameter $O(\rho/t)$, obtained by applying the effective inverse function theorem and a quantitative Lipschitz cell decomposition to $\Phi_k$. That diameter bound feeds into the coarea formula and yields the volume estimate $|E^{\delta,k}(x_1,r_1)\cap E^{\delta,k}(x_2,r_2)|\lesssim \log\delta^{-1}\,\delta^2/(\delta+|t_1-t_2|)$, and the summation argument of [6] converts the pair-wise volume bound into the multiplicity bound behind the discretised $L^2$ estimate.

What would settle it

For a fixed dimension $n$, compute the volume of the intersection $E^{\delta,k}(0,1)\cap E^{\delta,k}(t d_k,r)$ for pairs of ellipsoids whose parameters place them near the higher-order tangency set, and test whether the bound $\delta^2/(\delta+t)$ (up to a $\log\delta^{-1}$ factor) holds uniformly as $\delta\to 0$; a single sequence of configurations violating this bound would falsify Lemma 2.2 and with it the proof of the discretised $L^2$ estimate.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $n\geq 3$ and every $p>2$, the strong spherical maximal operator $M^{\mathrm{st}}$ is bounded on $L^p(\mathbb{R}^n)$, with a constant $C_{n,p}$ depending only on dimension and exponent. The paper proves this through the discretised $L^2$ bound (Theorem 1.3), which is new and independent of the earlier Fourier-analytic machinery: for every $\varepsilon>0$, the localised supremum over radii in a small cube satisfies $\|M^\delta f\|_{L^2}\lesssim_\varepsilon \delta^{-\varepsilon}\|f\|_{L^2}$. The discretised bound is used to obtain an $L^p$–Sobolev estimate for the local maximal operator, which is then interpolated against the $L^p$–Sobolev estimates of [10] and extended to unrestricted radii by a multiparameter frequency-localisation argument. In dimension three the necessary condition $p>2$ and the sufficiency $p>2$ coincide, so the strong spherical maximal conjecture is resolved there; in higher dimensions the conjecture remains open, and this paper leaves a gap between the proven range $p>2$ and the conjectured range $p>(n+1)/(n-1)$.

Load-bearing premise

The load-bearing assumption is the imported quantitative cell-decomposition theorem (Lemma 3.7): it asserts that the relevant real-algebraic pieces can be covered by $O(1)$ cells in which every pair of points can be joined by a curve of length comparable to their straight-line distance, with the comparability constant independent of the scale parameters; if that constant failed to be uniform, the diameter bound behind the volume estimate would collapse.

Editorial extensions

If this is right

The strong spherical maximal conjecture is settled in dimension three: the operator is bounded on $L^p(\mathbb{R}^3)$ for every $p>2$, exactly the range forced by the necessary condition.
In every dimension $n\geq 3$, the previously known boundedness range $p>2(n+1)/(n-1)$ is improved to $p>2$.
The discretised $L^2$ estimate (1.4) holds for all $n\geq 3$ with only a $\delta^{-\varepsilon}$ loss, for every $\varepsilon>0$.
The local maximal operator $M_{\mathrm{loc}}$ satisfies an $L^p$–Sobolev estimate with a positive power saving for every $p>2$, which is the input that makes the global frequency-localisation argument work.

Reading between the lines

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

If the same slicing construction is applied with ellipsoid centres confined to a $k$-dimensional subspace rather than a coordinate line, the argument suggests a hierarchy of multiparameter strong spherical estimates indexed by the dimension of the centre set; this is an extension the paper does not state.
The proof's universal-exceptional-set idea indicates that the $\log\delta^{-1}$ factor in the volume bound may be removable by choosing the slicing lines more carefully; testing this on the $n=3$ case would sharpen the constant without changing the main theorem.
The manuscript contains an anomalous inserted token '/suppress' before a cited theorem's name in the proof of Lemma 3.4; the surrounding argument still depends on that cited theorem for the connectivity of the semi-algebraic pieces, so the passage should be checked before relying on the quantitative constant.
A direct numerical evaluation of the volume bound in Lemma 2.2 on explicit pairs of ellipsoids would isolate the geometric mechanism from the Fourier-analytic interpolation and could indicate whether the $p>2$ range is improvable in higher dimensions.

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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

Short version: real advance, and the proof is largely sound. The main thing that worried me—uniformity of the Lipschitz constant in Pawlucki's cell decomposition—turns out not to be a problem.

The new content is the discretised incidence estimate (Theorem 1.3). The previous best range p>2(n+1)/(n-1) was due to Lee-Lee-Oh; getting down to p>2 for n≥3 is a genuine improvement and is sharp at n=3. The core of the proof, Lemma 2.2 plus Lemma 3.4, is a variable-slicing argument: fix a direction, remove the high-tangency set, then use a quantitative Jacobian bound and a dyadic coarea estimate. Lemma 3.3, which gives |det DωΦ| >~ t^{n-1} and |DωΦ^{-1}| ≤ C/t under a small Φ condition, is worked out explicitly and is believable. The covering-by-refinements step at the end of §3 is a neat way to handle the exceptional sets.

Soft spots: the Fourier-analytic reduction in §4 is mostly delegated to [10] and [17]. Probably fine for a harmonic analysis audience, but a referee should verify the interpolation and kernel domination arguments. Also, the paper imports, not reproves, the quantitative inverse function theorem (Lemma 3.5) and Pawlucki's cell decomposition (Lemma 3.7). That is acceptable, but it means the proof rests on two substantial external tools.

On the stress-test worry: Lemma 3.4 uses Lemma 3.7 to cover Ω_k(t,r;ρ) by O(1) sets of diameter O(ρ/t). The concern was that Pawlucki's L might depend on t,r. It does not. The semi-algebraic sets to which Lemma 3.7 is applied are images Φ_{t,r}(Ω∩U_m), and these have complexity bounded uniformly in t,r,ρ because the defining polynomials have degree O(1) and bounded coefficients. So Lemma 3.7 indeed gives L=O(1) uniformly, and the diameter estimate survives. That concern does not land.

The necessary-condition appendix is self-contained and useful.

Who is this for: harmonic analysts working on maximal operators and people interested in incidence-geometric methods in Fourier analysis. It deserves a serious referee. My recommendation: engage with it; likely outcome is acceptance after a careful check of the geometry in §3.

Referee Report

0 major / 6 minor

Summary. The paper proves L^p boundedness for p>2 of the strong spherical maximal operator M^st in R^n for all n≥3, improving the previous range p>2(n+1)/(n-1) obtained by Lee, Lee, and Oh and matching the conjectured sharp range when n=3. The proof combines a new discretised incidence-geometry estimate (Theorem 1.3) with the Fourier-analytic L^p-Sobolev estimates of Lee-Lee-Oh and a multiparameter Littlewood-Paley reduction. The core of the paper is a detailed proof of a volume bound (Lemma 2.2) for intersections of thin ellipsoidal annuli after removing exceptional sets of high-order tangencies, using a variable slicing argument, coarea estimates, and a quantitative inverse function theorem.

Significance. If correct, the result resolves the sharp range for the strong spherical maximal function in three dimensions and gives the best known range in all higher dimensions. The geometric part of the proof is substantial and carefully executed: the slicing and duality reductions, the exceptional-set removal, the coarea estimates, and the quantitative inverse function theorem are all developed in detail with explicit dimensional constants. The main novelty, the discretised incidence-geometry volume estimate, is likely to be useful for other multiparameter maximal problems. The proof does rely on one deep imported result, Pawlucki's quantitative Lipschitz cell decomposition (Lemma 3.7), which the authors should state with maximal precision concerning the dependence of the Lipschitz constant; they do, however, cite the theorem and use it in a way that appears consistent with its quantitative content.

minor comments (6)

[Lemma 3.4 / §3.2] In the proof of Lemma 3.4, the sets Φ_{t,r}(Ω_k(t,r;ρ)∩U_m) are not necessarily open because Ω_k includes the closed condition |ω_k|^3 ≥ 2c_n, whereas Lemma 3.7 is stated for open X; please add a sentence explaining that one applies the cell decomposition to the interior of these sets and absorbs the measure-zero boundary into the null set N.
[Lemma 3.7 / §3.2] Lemma 3.7 states that each cell is a regular L-cell for 'some L=O(1)' without specifying the dependence; since the argument in Lemma 3.4 requires L to be bounded by a constant depending only on n and the complexity of X (uniformly in t, r, and ρ), please state this dependence explicitly.
[§3.2, text near Lemma 3.5] The text contains the corrupted phrase 'A theorem of /suppress Lojasiewicz'; this should read 'A theorem of Łojasiewicz'.
[§2.1] In the paragraph after (2.1), 'a slight abuse of notion' should be 'a slight abuse of notation'.
[§4.2] The passage from the local operator M^loc to the global operator M^st is delegated entirely to [10, §3]; since this is a key step in the proof of Theorem 1.1, it would be helpful to include at least a sketch of the Littlewood-Paley argument and the modifications needed when invoking Proposition 4.1.
[References] Reference [19] (J. Thom) shares the title and page range of [13] (Milnor); please verify the citation and correct the author or title if needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Theorem 1.3 proof is self-contained modulo independent external real-algebraic-geometry results, and the Fourier-analytic interpolation uses Lee-Lee-Oh as an external input, not as the target conclusion.

full rationale

The claimed derivation does not reduce to its own inputs. Theorem 1.3, the main novelty, is proved via the discretised L2 bound from the volume estimate Lemma 2.2. Lemma 2.2 is proved directly by slicing, coarea, Lemma 3.1 (an independent algebraic-variety volume estimate from Wongkew), and Lemma 3.2. Lemma 3.2 in turn relies on Lemma 3.4, whose proof invokes Lemma 3.5 (Carbery-Ricci-Wright effective inverse function theorem) and Lemma 3.7, stated as Pawlucki's Lipschitz cell decomposition. These are external theorems with independent proofs; their hypotheses (semi-algebraic complexity, uniform Lipschitz constants) do not include the target maximal-function bound. The inverse-derivative uniformity needed for diameter O(ρ/t) is proved in Lemma 3.3 directly from the explicit polynomial map Φ_k. No parameter in the proof is fitted to the target Lp estimate, and no conclusion is renamed as a prediction. The final section interpolates Proposition 4.1 with [10, Proposition 2.2]; [10] proves p>2(n+1)/(n-1), a genuinely weaker input, and the interpolation plus Proposition 4.1 extends to p>2. Using the multiparameter Littlewood-Paley reduction from [10, §3] is a proof technique, not an import of Theorem 1.1. Self-citations to [8] and [22] are methodological only and are not load-bearing. The only nontrivial imported input, Pawlucki's quantitative cell decomposition, is a caveat for correctness if its uniformity fails, but importing it is not circularity because it is not equivalent to the target result and is not established by this paper's assumptions.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's central contribution is a geometric volume estimate proved from standard real algebraic geometry tools and prior Fourier-analytic estimates. It introduces no fitted parameters and no invented analytic or physical entities. The main external dependency is the quantitative cell-decomposition theorem of Pawlucki, which is cited rather than proved.

assumptions (8)

standard math Quantitative inverse function theorem for polynomial mappings (Carbery-Ricci-Wright [4, Lemma 3.2])
Used in Lemma 3.4 to decompose the preimage of Φ_k into O(1) diffeomorphic pieces and to control the diameter via the inverse derivative bound.
standard math Lipschitz cell decomposition in o-minimal structures (Pawlucki [14], Lemma 3.7)
Provides the quantitative regularity of the semi-algebraic pieces used to bound diam Ω_k^m by O(ρ/t). This is the weakest external input.
standard math Volume bound for tubular neighbourhoods of real algebraic varieties (Wongkew [21])
Gives the H^{n-2} measure bound in Lemma 3.1 for the intersection of two ellipsoids.
standard math Tarski-Seidenberg theorem and Milnor-Thom quantitative semi-algebraic geometry
Ensures the sets Φ_k(Ω∩U_m) have O(1) complexity and O(1) connected components, as needed in Lemma 3.4.
domain assumption Lp-Sobolev estimates for the strong spherical maximal operator for p > 2(n+1)/(n-1) (Lee-Lee-Oh [10, Proposition 2.2])
Imported as the interpolation input in Proposition 4.1; it is the previous best bound and is not reproved.
standard math Fourier multiplier kernel domination lemma of Schlag [17, Lemma 5.1]
Converts the discretized L2 bound into the L2-Sobolev estimate (4.1).
standard math Boundedness of the classical strong maximal function and multiparameter Littlewood-Paley theory
Used to pass from the local maximal operator with restricted radii to the global operator in the proof of Theorem 1.1.
standard math Coarea formula for Lipschitz maps (Federer [7])
Computes the volume of intersections of ellipsoidal annuli in Lemma 2.2.

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Pith. "Pith review of Improved $L^p$ bounds for the strong spherical maximal operator." pith.science (2026). https://pith.science/paper/VQSIQNGU

@misc{pith2026250202795,
  author       = {Pith},
  title        = {Pith review of: Improved $L^p$ bounds for the strong spherical maximal operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQSIQNGU}},
  note         = {Machine review of arXiv:2502.02795}
}

abstract

We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n \geq 3$. This matches the conjectured sharp range $p>(n+1)/(n-1)$ when $n=3$. For $n=2$ the analogous estimate was recently proved by Chen, Guo and Yang. Our result builds upon and improves an earlier bound of Lee, Lee and Oh. The main novelty is an estimate in discretised incidence geometry that bounds the volume of the intersection of thin neighbourhoods of axis-parallel ellipsoids. This estimate is then interpolated with the Fourier analytic $L^p$-Sobolev estimates of Lee, Lee and Oh.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Lacunary $\delta$-Discretised Spherical Maximal Operators
math.CA 2025-07 conditional novelty 6.0 of 10

Lacunary delta-discretised spherical maximal operators are Lp bounded for 1<p<infinity and H1 to weak L1, with constants independent of delta; the multi-parameter variant is also Lp bounded.

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Works this paper leans on

22 extracted references · 20 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [10]

$L^p$ bound on the strong spherical maximal function

Juyoung Lee, Sanghyuk Lee, and Sewook Oh, Lp bound on the strong spherical maximal function, to appear Proc. AMS, preprint: arXiv:2309.15308

work page Pith review arXiv

[2] [17]

Schlag, A generalization of Bourgain ’s circular maximal theorem , J

W. Schlag, A generalization of Bourgain ’s circular maximal theorem , J. Amer. Math. Soc. 10 (1997), no. 1, 103–122. MR 1388870

work page 1997

[3] [1]

S. Basu, R. Pollack, and M. F. Roy., Algorithms in real algebraic geometry (2nd ed.) , Springer- Verlag, Berlin, 2006

work page 2006

[4] [2]

Bochnak, M

J. Bochnak, M. Coste, and M. F. Roy., Real algebraic geometry , Springer, Berlin, 1998

work page 1998

[5] [3]

Edinburgh Math

Anthony Carbery, Covering lemmas revisited , Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 1, 145–150. MR 930022

work page 1988

[6] [4]

Anthony Carbery, Fulvio Ricci, and James W right, Maximal functions and singular integrals associated to polynomial mappings of Rn, Rev. Mat. Iberoamericana 19 (2003), no. 1, 1–22. MR 1993413 BOUNDS FOR THE STRONG SPHERICAL MAXIMAL OPERATOR 19

work page 2003

[7] [5]

Mingfeng Chen, Shaoming Guo, and Tongou Yang, A multi-parameter cinematic curvature , preprint: arXiv:2306.01606

arXiv

[8] [6]

Antonio Cordoba, The Kakeya maximal function and the spherical summation mul tipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 447949

work page 1977