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Maximal functions unify to one limit as dimension grows

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load-bearing objection Solid paper proving convergence of several maximal function norms to a universal limit in high dimensions the 3 major comments →

arxiv 2607.06041 v1 pith:LS6NWM6O submitted 2026-07-07 math.CA math.FAmath.MGmath.PR

High-dimensional limits and extremizers for maximal functions associated with log-concave densities

Valentina Ciccone , B{\l}a\.zej Wr\'obel This is my paper
classification math.CA math.FAmath.MGmath.PR MSC 42B2552A40
keywords maximal functionslog-concave densitieshigh-dimensional asymptoticsFourier multipliersthin-shell conjectureoperator normsconvex geometry
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 proves that as the spatial dimension d grows without bound, the L^p operator norms of several seemingly different maximal functions—the Hardy-Littlewood maximal function over balls, the Gaussian (heat semigroup) maximal function, the spherical maximal function, and indeed any maximal function associated with a radial log-concave density—all converge to a single universal quantity lambda(p). The authors establish explicit bounds: max{(2/5)(p/(p-1)), 1} <= lambda(p) <= p/(p-1). They further prove an extremality result: among all symmetric convex bodies in high dimensions, the Euclidean ball achieves the asymptotically minimal L^p operator norm for its associated maximal function. The central mechanism is a transference principle that controls maximal functions through their Fourier multiplier symbols, combined with variance and thin-shell concentration estimates for log-concave random vectors imported from high-dimensional convex geometry.

Core claim

The key discovery is that the Fourier multiplier symbols of all these different averaging operators become pointwise indistinguishable from the Gaussian multiplier at a rate of d^{-1/4+epsilon} as the dimension d grows. This is a consequence of the thin-shell phenomenon: isotropic log-concave random vectors concentrate on a thin spherical shell of radius proportional to sqrt(d), making their spherical averages behave like Gaussian averages in the Fourier domain. The transference principle (Theorem 1.8) then converts this pointwise symbol closeness into an operator norm bound: if two multiplier symbols are uniformly close, the corresponding maximal operators have close L^p norms. Since the L^

What carries the argument

The transference principle (Theorem 1.8) bounds the L^2 operator norm of a maximal function by the L^infinity norm of its Fourier multiplier symbol to the power 1/4, provided the symbol satisfies certain uniform pointwise bounds. This is combined with Proposition 3.1, which uses the Klartag-Lehec variance bound (Theorem 1.9) to show that the Fourier multiplier of any spherical average of a log-concave density is within C*d^{-1/4+epsilon} of the Gaussian multiplier. Monotonicity of the Gaussian maximal function norm in d (Theorem 1.6, proved via tensor product structure) ensures the limit lambda(p) exists.

Load-bearing premise

The transference principle requires that the Fourier multiplier symbols satisfy certain pointwise bounds uniformly in the dimension d, and the symbol approximation relies on the Klartag-Lehec variance bound for general log-concave measures. If these uniform-in-d estimates were to fail for some class of log-concave densities, the convergence to lambda(p) would not hold for that class.

What would settle it

If one could exhibit a sequence of log-concave densities in growing dimension whose Fourier multiplier symbols do not satisfy the uniform bounds (1.16)-(1.17), or for which the variance bound (1.20) fails, the convergence to the universal limit would break down for that sequence.

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

If this is right

  • The exact value of lambda(p) remains open; the paper shows it is between max{(2/5)(p/(p-1)), 1} and p/(p-1), and the authors note they do not know whether lambda(p) equals p/(p-1).
  • The cube maximal function Q* also satisfies lambda(p) <= lim_{d->inf} ||Q*||_{L^p}, so the ball is asymptotically minimal not just among log-concave densities but also compared to cubes.
  • The paper's Question (1.27) asks whether all these maximal operators have norm 1 when restricted to radial input functions, which the three test examples in Section 7 suggest but do not prove.
  • The framework extends to the Dosidis-Grafakos family S_{alpha,*} connecting ball and spherical maximal functions, all of which converge to lambda(p) as well.

Where Pith is reading between the lines

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

  • If the exact value of lambda(p) were determined, it would simultaneously pin down the asymptotic operator norm for ball, Gaussian, and spherical maximal functions—a unified answer to what have been treated as separate problems.
  • The reliance on the thin-shell conjecture resolution suggests a deeper structural connection: high-dimensional concentration phenomena in convex geometry may systematically govern the asymptotic behavior of harmonic analysis operators.
  • The transference principle could potentially be applied to other averaging operators whose Fourier multipliers satisfy the required uniform bounds, extending the universality phenomenon beyond the classes studied here.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 10 minor

Summary. The paper establishes that the L^p(R^d) operator norms of maximal functions associated with radial log-concave densities, the Hardy-Littlewood maximal function over balls, the Gaussian maximal function, and the spherical maximal function all converge to a single universal limit λ(p) as d→∞. The authors prove quantitative bounds max{(2/5)(p/(p-1)), 1} ≤ λ(p) ≤ p/(p-1) and show an extremality property: among all symmetric convex bodies, the ball achieves the asymptotically minimal L^p operator norm. The proof proceeds via a transference principle (Theorem 1.8, §2) that controls maximal functions through Fourier multiplier symbols, combined with pointwise estimates on these symbols (Propositions 3.1 and 5.1) that rely on the Klartag-Lehec variance bound (Theorem 1.9) and Gaussian thin-shell concentration. Dimension-free L^q bounds and Marcinkiewicz interpolation extend the L^2 estimates to all p∈(1,∞). The existence of λ(p) follows from monotonicity of the Gaussian maximal function norm in d (Theorem 1.6, §6).

Significance. The paper makes a notable contribution by connecting high-dimensional convex geometry (variance and thin-shell bounds) to the study of maximal functions, a direction that has been surprisingly underexplored. The universality of the limit λ(p) across multiple classical maximal operators is a clean and appealing result. The lower bound λ(p) ≥ (2/5)(p/(p-1)) is explicit and non-trivial. The transference principle (Theorem 1.8) is a useful technical tool, and the verification of its hypotheses for both log-concave densities (Lemma 3.3) and the spherical multiplier (Proposition 5.3, Appendix A) is carried out carefully. The extremality result (Theorem 1.4) is a satisfying consequence. The paper also provides falsifiable quantitative decay rates (d^{-α_p} with α_p < min(2/p, 2(1-1/p))/16) and explicitly notes (Remark 1.10) that any power-type variance bound suffices, so the main results do not depend on the optimality of the Klartag-Lehec theorem.

major comments (3)
  1. §3.1, equation (3.9): The claim that sup_{d≥1} ||(A_K - G_σ)*||_{L^q(R^d)} < ∞ for all q∈(1,∞] is too strong as stated. For q∈(1,2] and small d, the spherical maximal function S* is bounded on L^q(R^d) only for q > d/(d-1), which fails at d=2 for q∈(1,2]. Since the uniform L^q bound for A_K* is obtained via the pointwise bound A_K* f ≤ S* f, the claimed uniformity in d cannot hold for all q∈(1,∞] and all d≥1. This issue is correctly handled in §5.1 (equation (5.9)) for the spherical case, where the supremum is taken over d ≥ d_0(q) := ⌊q/(q-1)⌋ + 1. The same fix should be applied in (3.9). This does not affect the main results, since the interpolation in §3.1 only requires the L^q bound for sufficiently large d, but the statement as written is incorrect.
  2. §3, proof of Proposition 3.1 (equation (3.5) and surrounding text): In the decomposition of the expectation over the events ||X| - √d| ≥ d^ε and ||X| - √d| ≤ d^ε, the Gaussian tail is estimated using P(|X| - √d| ≥ d^ε) ≲ e^{-c d^{ε/2}} (citing [23, Corollary 1.3]). However, the standard Laurent-Massart bound (as used later in Proposition 5.1, equation (5.4)) gives P(||X|² - d| ≥ 4d^{1/2+ε}) ≤ e^{-d^ε}, which translates to P(||X| - √d| ≥ ~d^{ε-1/2}) ≤ e^{-d^ε}. The threshold d^ε on ||X| - √d| (rather than on ||X|² - d|) appears to give a weaker tail bound e^{-c d^{ε/2}} rather than e^{-d^ε}. The authors should clarify which concentration inequality is being applied and verify that the resulting bound e^{-c d^{ε/2}} is sufficient for the final conclusion (it appears to be, since any super-polynomial decay suffices, but the discrepancy with the sharper bound used in Proposition 5.1 shouldbe
  3. §6, proof of Proposition 6.1: The monotonicity argument embeds f∈L^p(R^d) into L^p(R^{d+1}) via tensor product with φ_N(s) = |s|^{-1/p} 1_{[1,N]}(|s|). The key step shows that G_t^1(φ_N)(s) ≥ φ_{N/2}(s) · 1_{[10TM,∞)}(|s|) · I(M) for 0 < t < T, where I(M) → 1 as M → ∞. The argument requires N > 20TM and then takes N → ∞ followed by M → ∞ and T → ∞. The order of limits is important: for each fixed T and M, N is sent to infinity, and the ratio of L^p norms of φ_{N/2}·1_{[10TM,∞)} and φ_N converges to I(M). The authors should verify that the implicit constants in the convexity argument (the inequality (φ_N(s-ty) + φ_N(s+ty))/2 ≥ φ(s) for |s| > 10TM) are indeed independent of t ∈ (0,T) and y ∈ [0,M], as this is load-bearing for the uniformity needed before taking the supremum over t.
minor comments (10)
  1. Abstract and Theorem 1.1: The lower bound is stated as (2/5)(p/(p-1)) but Theorem 1.6 and Lemma 6.4 establish that the constant c > 0.4, giving (2/5)(p/(p-1)) as a clean lower bound. The abstract should perhaps note that this specific constant comes from the infimum computation in Lemma 6.4, to avoid confusion about its provenance.
  2. §1.1.1, page 6: 'conlcude' should be 'conclude' (appears twice, once in the Stein-Strömberg discussion and once in the semigroup discussion on page 6).
  3. §2, page 10: In the chain of inequalities leading to (2.3), the constant 32√π appears, but the intermediate step shows 8√π followed by 16√π. The authors should verify the numerical constants are consistent, though the exact values are not load-bearing.
  4. §3, Lemma 3.2: The identity (3.2) expresses a(ξ) as an integral over K(y) of an expectation over X~N(0,I_d). The normalization should be clarified: the expectation E_{X~N(0,I_d)}[e^{-2πi⟨|y|·X/|X|, ξ⟩}] involves X/|X| which is uniform on S^{d-1}, but the identity as written may need a factor involving the surface measure. The proof seems correct but the normalization could be stated more explicitly.
  5. §5, Proposition 5.3, page 17: In the case |ξ| ≥ d, the bound |μ(ξ)| ≲ 1/|ξ| is derived using |J_ν(r)| ≤ r^{-1/2} for r ≥ 2ν (citing [25, Theorem 3]). The condition r ≥ 2ν translates to 2π|ξ| ≥ d-2, which is satisfied when |ξ| ≥ d for d ≥ 3. This should be stated explicitly for clarity.
  6. §7, Lemma 7.1: The statement says s_{p,d} = 1 for d ≥ max{3, d'(p)}, where d'(p) is the smallest d with p ≥ d/(d-2). The superharmonicity argument is clean, but the reference to [14, p. 478] for the suggestion that b_{p,d} = 1 for p ≥ d/(d-2) could be made more precise with a page or equation number.
  7. Appendix A, Lemma A.2: The contour deformation argument follows [33, Lemma 4.1] with minor modifications. The bound (A.5) involves a term e^{-d/10}/√d, and the constant 1/10 appears to come from (3/4)^{(d-3)/2} ≈ e^{-d/10}. This should be stated explicitly for the reader's convenience.
  8. References: [23] (Klartag-Lehec, 'Thin-shell bounds via parallel coupling') is listed as arXiv 2026. The authors should verify whether a published version is available and update the reference accordingly.
  9. §1.1.4, page 7: The text states 'A first highly non-trivial bound in the general case was proved in [21], showing that (1.19) holds with δ_d ≤ C√(d/log d).' The reference [21] is Klartag 2007, which is correct, but the bound δ_d ≤ C√(d/log d) is a well-known result that could also cite the earlier work of Klartag (2006) for context.
  10. Notation (page 9, item 5): The Poisson projections S_n = P_{2^{n-1}} - P_{2^n} are defined for n∈Z, but the resolution of the identity f = Σ_{n∈Z} S_n f should perhaps note that this is an L^2 decomposition, which is mentioned but could be more prominent.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying three points requiring attention. All three are legitimate; two require straightforward corrections and one requires a clarification with minor verification. None affects the main results.

read point-by-point responses

  1. Referee: §3.1, equation (3.9): The claim that sup_{d≥1} ||(A_K - G_σ)*||_{L^q(R^d)} < ∞ for all q∈(1,∞] is too strong as stated, because for q∈(1,2] and small d the spherical maximal function S* is bounded on L^q only for q > d/(d-1), which fails at d=2 for q∈(1,2]. The same fix as in §5.1 (equation (5.9)) should be applied.

    Authors: The referee is entirely correct. The uniform L^q bound in (3.9) as stated cannot hold for all d≥1 when q∈(1,2], since A_K* f ≤ S* f and S* is unbounded on L^q(R^d) for q ≤ d/(d-1). The fix is straightforward: we will replace the supremum over d≥1 in (3.9) with a supremum over d ≥ d_0(q) := ⌊q/(q-1)⌋ + 1, exactly as is already done in (5.9) for the spherical case. This is consistent with the rest of the argument, since the Marcinkiewicz interpolation in §3.1 only requires the L^q bound for sufficiently large d (the L^2 bound (3.8) provides the decay estimate, and interpolation is applied for d large enough that both estimates hold). We will also add a brief sentence explaining why restricting to d ≥ d_0(q) is sufficient for the interpolation argument. revision: yes

  2. Referee: §3, proof of Proposition 3.1 (equation (3.5) and surrounding text): The Gaussian tail is estimated using P(||X| - √d| ≥ d^ε) ≲ e^{-c d^{ε/2}}, but the standard Laurent-Massart bound gives P(||X|² - d| ≥ 4d^{1/2+ε}) ≤ e^{-d^ε}, which translates to P(||X| - √d| ≥ ~d^{ε-1/2}) ≤ e^{-d^ε}. The threshold d^ε on ||X| - √d| (rather than on ||X|² - d|) appears to give a weaker tail. Clarify which concentration inequality is applied and verify sufficiency.

    Authors: The referee has correctly identified a discrepancy between the concentration inequality applied in Proposition 3.1 and the sharper bound used in Proposition 5.1. In Proposition 3.1, we apply [23, Corollary 1.3] in the form P(||X| - √d| ≥ t) ≲ e^{-c√t} with t = d^ε, yielding e^{-c d^{ε/2}}. This is indeed weaker than what one obtains by applying the Laurent-Massart bound (as done in Proposition 5.1) to ||X|² - d| directly. The reason for the discrepancy is that in Proposition 3.1 we work with the threshold on ||X| - √d| rather than on ||X|² - d|, and the conversion introduces the square root in the exponent. However, as the referee notes, any super-polynomial decay suffices for the conclusion: the tail term e^{-c d^{ε/2}} is absorbed into the d^{-1/4+ε} bound (with ε → 0⁺) that Proposition 3.1 establishes. We will clarify in the revised text which form of the concentration inequality is being applied and why the resulting bound, while not sharp, is sufficient. We will also add a remark noting that the sharper Laurent-Massart bound used in Proposition 5.1 could equally well be applied here, but is not necessary. revision: partial

  3. Referee: §6, proof of Proposition 6.1: The order of limits (N→∞, then M→∞, then T→∞) and the uniformity of implicit constants in the convexity argument (φ_N(s-ty) + φ_N(s+ty))/2 ≥ φ(s) for |s| > 10TM, independent of t∈(0,T) and y∈[0,M], should be verified.

    Authors: We have carefully re-examined the convexity argument and confirm that the implicit constants are indeed independent of t ∈ (0,T) and y ∈ [0,M], as the referee requests. The key observation is that φ(s) = |s|^{-1/p} is convex on each of (-∞,0) and (0,∞). For |s| > 10TM and y ∈ [0,M], t ∈ (0,T), we have |s ± ty| ≥ |s| - tM > |s| - TM > 9TM > 0. In particular, s - ty and s + ty have the same sign as s (since |ty| < TM < |s|), so both lie in the same convexity region as s. The convexity inequality (φ_N(s-ty) + φ_N(s+ty))/2 ≥ φ((s-ty+s+ty)/2) = φ(s) then holds with no dependence on t or y beyond the constraint that |s| > 10TM ensures the arguments remain in the same half-line. Moreover, for |s| ∈ (10TM, N/2), both |s ± ty| lie in [1, N] (since |s| - TM > 9TM ≥ 9 > 1 when M ≥ 1, and |s| + TM < N/2 + TM < N when N > 20TM), so φ_N coincides with φ at these points. The factor I(M) = (2/√(2π))∫_0^M e^{-y²/2} dy is also independent of t. Thus the ratio of L^p norms converges to I(M) as N → ∞ for each fixed T, M, uniformly in t ∈ (0,T), which is what is needed before taking the supremum over t. We will add a clarifying sentence in the revised proof making the sign-preservation argument explicit, so that the uniformity is transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity found. The derivation chain is self-contained against external benchmarks.

full rationale

The paper's central results are derived from genuinely external inputs. The variance bound (Theorem 1.9) is restated from Klartag-Lehec [23] (independent authors). The thin-shell Gaussian concentration comes from Laurent-Massart [28] (independent). Bessel function estimates come from Krasikov [25] (independent). The transference principle (Theorem 1.8) is proved from scratch in Section 2, with the method attributed to multiple independent sources ([6, 10, 15, 35]) in addition to [9] where Wróbel is a co-author. Proposition 3.1's proof is self-contained given the external variance bound. Proposition 5.3's Bessel function estimates are proved in detail in Appendix A, with [33] (Mirek-Szarek-Wróbel) cited for the method but the paper explicitly providing the missing gradient proof. The self-citations to [9] and [33] are not load-bearing in a circular sense: the proofs are reproduced or independently verified in the paper, and the underlying methods trace to multiple independent authors. The universal limit λ(p) is defined as lim ||G*_d||, an independently defined quantity, and the convergence of other maximal function norms to it is a non-trivial consequence of the approximation results, not a definition disguised as a prediction. No step in the derivation chain reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are introduced: all constants are universal (independent of d, K, p). No new entities are postulated. The axioms are all external results from high-dimensional convex geometry (Klartag-Lehec), probability (Laurent-Massart), and harmonic analysis (Bourgain, Carbery, Deleaval-Guédon-Maurey). The paper is a derivation from these external inputs, not a fitting exercise.

axioms (5)
  • domain assumption Variance type bound for isotropic log-concave random vectors: E(|X| - σ√d)² ≤ Cσ² (Theorem 1.9, from Klartag-Lehec [23])
    Load-bearing for Proposition 3.1, which controls |a(ξ) - g_σ(ξ)|. Without this, the uniform-in-d multiplier estimate fails. The authors note any power-type improvement over trivial δ_d = O(√d) would suffice (Remark 1.10).
  • domain assumption Thin-shell concentration for Gaussian random vectors: P(||X|² - d| ≥ 4d^{1/2+ε}) ≤ e^{-d^ε} (from Laurent-Massart [28, Lemma 1])
    Used in Proposition 5.1 to control the spherical multiplier symbol μ(ξ) near the Gaussian symbol. Also used implicitly in Proposition 3.1 via [23, Corollary 1.3].
  • domain assumption Dimension-free L^q estimates for maximal functions associated with log-concave densities and spherical means (from [12], [9])
    Required for the Marcinkiewicz interpolation step in §3.1 and §5.1, extending L^2 estimates to all p ∈ (1,∞).
  • domain assumption Pointwise multiplier estimates for log-concave densities: |k̂(ξ) - 1| ≤ 2πσ|ξ|, |k̂(ξ)| ≤ 1/(2πσ|ξ|), |⟨ξ,∇k̂(ξ)⟩| ≤ 2 (from [12, Lemma 5.10])
    Used in Lemma 3.3 to verify the hypotheses of the transference principle (Theorem 1.8) for the spherical average A_K.
  • standard math Bessel function decay estimates: |J_ν(r)| ≤ r^{-1/2} for ν ≥ 1/2, r ≥ 2ν (from Krasikov [25, Theorem 3])
    Used in Proposition 5.3 to verify the transference principle hypotheses for the spherical multiplier μ.

pith-pipeline@v1.1.0-glm · 34943 in / 3387 out tokens · 591610 ms · 2026-07-08T18:20:24.567335+00:00 · methodology

0 comments
read the original abstract

We introduce a unified framework to establish the high-dimensional asymptotic behavior of maximal functions associated with radial log-concave probability densities, encompassing the maximal heat semigroup, Hardy-Littlewood maximal function over Euclidean balls, and, additionally, maximal spherical means. Namely, for any $p \in (1, \infty)$, we prove that the $L^p(\mathbb{R}^d)$ operator norms of these maximal operators all converge as the dimension $d \to \infty$ to a single, universal limit $\lambda(p)$. Furthermore, by proving that the $L^p$ operator norms for the heat semigroup $\mathcal G_*^d$ are monotonically non-decreasing in the dimension, we provide explicit quantitative bounds on the universal limit, showing that $\frac{2}{5}\frac{p}{p-1} \le \|\mathcal{G}_*^1\|_{L^p(\mathbb{R}) \to L^p(\mathbb{R})} \le \lambda(p) \le \frac{p}{p-1}$. We also prove an extremality property: among all symmetric convex bodies in high dimensions, the maximal operator associated with the Euclidean ball achieves the asymptotically minimal $L^p$ operator norm. Our main results are established via a general transference principle that allows us to control maximal functions via Fourier multiplier symbols. To estimate these symbols uniformly across log-concave densities, we import variance type bounds and thin-shell type concentration of measure results, which are novel tools in the study of maximal functions. In particular, to prove the extremality property, we require a variance type bound for general log concave measures established in a recent series of breakthroughs in high dimensional convex geometry.

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