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arxiv: 2511.11283 · v2 · submitted 2025-11-14 · 🧮 math.CA

Problems on spherical maximal functions

Joris Roos , Andreas Seeger This is my paper

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classification 🧮 math.CA
keywords spherical maximal functionsfractal dilation setsharmonic analysismaximal operatorsconjecturessurveyaveraging operators
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The pith

Spherical maximal functions face open conjectures when dilations form fractal sets.

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

The paper surveys old and new conjectures and results on spherical maximal functions, with special focus on cases where the dilation set is fractal. These operators involve taking the supremum of averages over spheres whose radii belong to a given set. A sympathetic reader would care because the boundedness and pointwise convergence properties of such operators connect directly to differentiation of integrals and other core questions in harmonic analysis. Clarifying the fractal cases would determine the reach of averaging methods in Euclidean spaces of various dimensions.

Core claim

The authors survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

What carries the argument

Spherical maximal functions, which take the supremum of spherical averages over a set of radii that may be fractal, and whose boundedness properties depend on the geometry of that set.

If this is right

  • Boundedness of the maximal operator holds or fails according to the dimension and the Hausdorff dimension of the fractal dilation set.
  • Pointwise convergence of spherical means occurs almost everywhere under restrictions on the fractal properties of the radius set.
  • The survey organizes results that link geometric properties of the dilation set to analytic behavior of the operators.

Where Pith is reading between the lines

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

  • Resolving the surveyed conjectures could inform similar questions about maximal averages over other curved surfaces or manifolds.
  • Numerical checks on specific fractal sets such as Cantor sets of radii might offer evidence supporting or challenging the stated open problems.
  • The emphasis on fractal dilations suggests that techniques from geometric measure theory could be adapted to settle some of the listed conjectures.

Load-bearing premise

The survey accurately represents the current state of conjectures and results in the literature without significant omissions or misstatements.

What would settle it

A new theorem or counterexample resolving one of the emphasized open conjectures on fractal spherical maximal functions would test whether the survey's account of the open problems is complete and accurate.

Figures

Figures reproduced from arXiv: 2511.11283 by Andreas Seeger, Joris Roos.

Figure 1
Figure 1. Figure 1: illustrates the situation in the theorem. Within the critical triangle spanned by the points Q4,γ, Q4,β, Q3,β, the boundary of TE may follow an arbitrary convex curve segment. 1 q 1 p Q1 Q2,β Q3,β Q4,γ 1 q 1 p Q1 Q2,β Q3,β Q4,γ Q4,β [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Building a type set. Outline of proof for the inclusion Q(β, γ) ⊂ TE. One analyzes the L p → L q bounds for the maximal operator M j E in (3.5). A straightforward O(2j ) bound for the convolution kernel σj yields (5.4) ∥M j E ∥L1→L∞≲2 j [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
read the original abstract

We survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript surveys old and new conjectures and results on various types of spherical maximal functions, with emphasis on problems involving fractal dilation sets.

Significance. If the survey accurately captures the literature, it offers a useful compilation of open problems in harmonic analysis. Highlighting fractal dilation sets may direct attention to under-explored variants of the spherical maximal operator and related questions.

minor comments (2)
  1. [Abstract/Introduction] The abstract and introduction should explicitly delineate the scope of the survey (e.g., which classes of spherical maximal functions are included or excluded) to help readers evaluate completeness.
  2. Ensure that citations to recent preprints or updates on the listed conjectures are current, as the field evolves quickly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and positive recommendation of minor revision. The assessment that the survey compiles open problems in harmonic analysis and highlights fractal dilation sets is appreciated. No specific major comments were provided in the report, so we interpret the minor revision as a general request to ensure accuracy and clarity in the literature survey.

Circularity Check

0 steps flagged

No significant circularity in survey of conjectures

full rationale

The paper is a survey that collects and highlights existing conjectures and results on spherical maximal functions with fractal dilation sets, without presenting any new derivations, proofs, parameter fittings, or predictions. No equations or load-bearing claims appear that could reduce to self-citations or inputs by construction. The content draws from external cited literature in a standard descriptive manner, making the paper self-contained against external benchmarks with no internal circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a survey drawing on standard results in harmonic analysis and does not introduce new free parameters, axioms beyond background math, or invented entities.

axioms (1)
  • standard math Standard boundedness results and techniques for maximal functions in harmonic analysis.
    The survey relies on established theorems from the field for context and comparison.

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    math.AP 2026-04 unverdicted novelty 5.0

    The fractional spherical maximal function and its lacunary counterpart satisfy restricted weak type estimates at the boundary of their L^p-L^q boundedness regions.

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