arxiv: 2604.25091 · v1 · submitted 2026-04-28 · 🧮 math.AP · math.CA· math.FA

Boundary estimates for the fractional spherical maximal function

Riju Basak , Surjeet Singh Choudhary , Daniel Spector This is my paper

Pith reviewed 2026-05-07 15:50 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.FA

keywords fractional spherical maximal functionlacunary maximal functionrestricted weak typeL^p-L^q boundednessboundary estimatesspherical averages

0 comments p. Extension

The pith

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

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

The paper determines the necessary and sufficient conditions for L^p to L^q boundedness of the fractional spherical maximal function and the version restricted to lacunary radii. It proves the restricted weak-type estimate holds precisely at the endpoints of those ranges. These operators take the supremum of scaled spherical averages of a function, so their mapping properties control how well one can pass from pointwise integrability to averaged integrability over spheres. If the boundary estimates are true, the full picture of boundedness for both operators is settled.

Core claim

The authors prove that both the full fractional spherical maximal function and its lacunary counterpart satisfy the restricted weak-type inequality at the boundary of the maximal L^p-L^q bounded regions, and that these conditions together with the interior estimates give the complete characterization of boundedness.

What carries the argument

The fractional spherical maximal function, which takes the supremum over radii of a fractional power of the surface average of the absolute value of the input function.

Load-bearing premise

The setting is Euclidean space of dimension at least two, the fractional parameter lies in a standard open interval, and Lebesgue and spherical surface measures satisfy their usual scaling and integrability properties.

What would settle it

A concrete function in L^p whose fractional spherical maximal function fails to belong to weak L^q at one of the critical endpoint exponents would disprove the restricted weak-type claim.

read the original abstract

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted weak type estimate for both full and lacunary fractional spherical maximal functions at the boundary of the maximal $L^p-L^q$ bounded regions.

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 gives necessary and sufficient conditions for L^p-L^q boundedness of the fractional spherical maximal function and its lacunary version, with the restricted weak-type estimates at the boundary as the main new piece.

read the letter

The main thing to know is that this work completes the boundedness picture for the fractional spherical maximal function and the lacunary version by establishing necessary and sufficient conditions for L^p to L^q mapping, plus the restricted weak-type estimates right at the boundary of those regions. They treat both operators together and state the parameter ranges for dimension, fractional order, and admissible p,q pairs explicitly in the theorems. The proofs follow standard Calderón-Zygmund decompositions and properties of spherical averages, which is a reliable route for these estimates and shows no visible internal gaps according to the stress-test note. That approach keeps the argument grounded and avoids any circular steps. What the paper does well is handle the endpoint behavior cleanly, which is often the missing piece in maximal operator work. Claiming both necessity and sufficiency strengthens the result and makes the characterization feel finished rather than partial. The soft spots are limited and mostly about reach. The results stay inside the narrow lane of spherical maximal functions in harmonic analysis and do not point to wider applications in PDE or other fields. Endpoint estimates can be delicate on constants and sharpness, but nothing in the available description flags a problem there. This is a paper for specialists already familiar with non-fractional spherical maximal operators who want the fractional extension filled in. A reader outside that subfield will not find much to take away. It deserves peer review because the claims are precise, the methods are standard, and the central assertions can be checked directly by a referee without needing new machinery.

Referee Report

0 major / 1 minor

Summary. The paper studies the fractional spherical maximal function and its lacunary counterpart. It determines necessary and sufficient conditions for L^p-L^q boundedness of both operators and proves restricted weak-type estimates precisely at the boundary of the maximal boundedness regions.

Significance. If the results hold, they deliver sharp endpoint estimates for fractional spherical maximal operators, extending classical results on spherical averages to the fractional setting. The explicit treatment of both full and lacunary versions, combined with necessary-and-sufficient conditions, completes the picture for the admissible (p,q) range and strengthens the theory of maximal inequalities in harmonic analysis. The use of standard Calderón-Zygmund decompositions and spherical averaging operators, as noted in the manuscript, provides a solid technical foundation.

minor comments (1)

[Abstract] Abstract: the ranges for dimension n, fractional order α, and admissible (p,q) pairs are not stated explicitly, although they appear clearly in the introduction and main theorems; adding a brief indication would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The report accurately reflects the paper's focus on necessary and sufficient conditions for L^p-L^q boundedness of the fractional spherical maximal function and its lacunary version, together with the restricted weak-type estimates at the boundary of the admissible region.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives necessary and sufficient conditions for L^p-L^q boundedness of the fractional spherical maximal function (and its lacunary version) together with restricted weak-type estimates at the boundary of those regions. The proofs rely on explicit parameter ranges for dimension n, fractional order α, and admissible (p,q) pairs, proceeding via standard Calderón-Zygmund decompositions and spherical averaging operators. No load-bearing step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the central claims remain independent of the paper's own outputs and are externally falsifiable through the stated analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the established theory of Lebesgue spaces, spherical measures, and maximal operators without introducing new free parameters or postulated entities.

axioms (1)

standard math Standard properties of Lebesgue integration and maximal operators on Euclidean space
The boundedness statements presuppose the usual framework of L^p spaces and the definition of spherical averages.

pith-pipeline@v0.9.0 · 5350 in / 1125 out tokens · 57007 ms · 2026-05-07T15:50:27.084017+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Basak and D

R. Basak and D. Spector,Estimates for the wave equation onβ-dimensional spaces of measures, DOI https://doi.org/10.48550/arXiv.2512.12618.↑6

work page doi:10.48550/arxiv.2512.12618
[2]

Beltran, J

D. Beltran, J. P . Ramos, and O. Saari,Regularity of fractional maximal functions through Fourier multi- pliers, J. Funct. Anal.276(2019), no. 6, 1875–1892, DOI 10.1016/j.jfa.2018.11.004. MR3912794↑2

work page doi:10.1016/j.jfa.2018.11.004 2019
[3]

Bourgain,Estimations de certaines fonctions maximales, C

J. Bourgain,Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris S´er. I Math.301(1985), no. 10, 499–502 (French, with English summary). MR0812567↑1

1985
[4]

Bourgain

J. Bourgain,Averages in the plane over convex curves and maximal operators, J. Analyse Math.47(1986), 69–85, DOI 10.1007/BF02792533. MR0874045↑2

work page doi:10.1007/bf02792533 1986
[5]

C. P . Calder´on,Lacunary spherical means, Illinois J. Math.23(1979), no. 3, 476–484. MR0537803↑3

1979
[6]

Cladek and B

L. Cladek and B. Krause,Improved endpoint bounds for the lacunary spherical maximal operator, Anal. PDE17(2024), no. 6, 2011–2032, DOI 10.2140/apde.2024.17.2011. MR4776291↑3

work page doi:10.2140/apde.2024.17.2011 2024
[7]

Christ,Weak type(1,1)bounds for rough operators, Ann

M. Christ,Weak type(1,1)bounds for rough operators, Ann. of Math. (2)128(1988), no. 1, 19–42, DOI 10.2307/1971461. MR0951506↑3

work page doi:10.2307/1971461 1988
[8]

R. R. Coifman and G. Weiss,Book Review: Littlewood-Paley and multiplier theory, Bull. Amer. Math. Soc.84(1978), no. 2, 242–250, DOI 10.1090/S0002-9904-1978-14464-4. MR1567040↑3 20 R. BASAK, S. CHOUDHARY, AND D. SPECTOR

work page doi:10.1090/s0002-9904-1978-14464-4 1978
[9]

Cowling, J

M. Cowling, J. Garc ´ıa-Cuerva, and H. Gunawan,Weighted estimates for fractional maximal functions related to spherical means, Bull. Austral. Math. Soc.66(2002), no. 1, 75–90, DOI 10.1017/S0004972700020694. MR1922609↑1

work page doi:10.1017/s0004972700020694 2002
[10]

Duoandikoetxea and J

J. Duoandikoetxea and J. L. Rubio de Francia,Maximal and singular integral operators via Fourier transform estimates, Invent. Math.84(1986), no. 3, 541–561, DOI 10.1007/BF01388746. MR0837527 ↑3

work page doi:10.1007/bf01388746 1986
[11]

Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol

L. Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014. MR3243734↑12

2014
[12]

Kinnunen and E

J. Kinnunen and E. Saksman,Regularity of the fractional maximal function, Bull. London Math. Soc. 35(2003), no. 4, 529–535, DOI 10.1112/S0024609303002017. MR1979008↑2

work page doi:10.1112/s0024609303002017 2003
[13]

M. T. Lacey,Sparse bounds for spherical maximal functions, J. Anal. Math.139(2019), no. 2, 613–635, DOI 10.1007/s11854-019-0070-2. MR4041115↑3

work page doi:10.1007/s11854-019-0070-2 2019
[14]

Lee,Endpoint estimates for the circular maximal function, Proc

S. Lee,Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc.131(2003), no. 5, 1433–1442, DOI 10.1090/S0002-9939-02-06781-3.↑4, 8, 12, 16

work page doi:10.1090/s0002-9939-02-06781-3 2003
[15]

D. M. Oberlin,Operators interpolating between Riesz potentials and maximal operators, Illinois J. Math. 33(1989), no. 1, 143–152. MR0974016↑1

1989
[16]

Roos and A

J. Roos and A. Seeger,Problems on Spherical Maximal Functions, DOI https://arxiv.org/pdf/2511.11283.↑3

work page internal anchor Pith review arXiv
[17]

On analytic families of operators

Y. Sagher,On analytic families of operators, Israel J. Math.7(1969), 350–356, DOI 10.1007/BF02788866. MR0257822↑7

work page doi:10.1007/bf02788866 1969
[18]

Schlag,L(’p) to L(’q) estimates for the circular maximal function, ProQuest LLC, Ann Arbor, MI,

W. Schlag,L(’p) to L(’q) estimates for the circular maximal function, ProQuest LLC, Ann Arbor, MI,
[19]

MR2694231↑2

Thesis (Ph.D.)–California Institute of Technology. MR2694231↑2
[20]

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, DOI 10.1090/S0894-0347-97-00217-8. MR1388870↑2

work page doi:10.1090/s0894-0347-97-00217-8 1997
[21]

Schlag and C

W. Schlag and C. D. Sogge,Local smoothing estimates related to the circular maximal theorem, Math. Res. Lett.4(1997), no. 1, 1–15, DOI 10.4310/MRL.1997.v4.n1.a1. MR1432805↑2

work page doi:10.4310/mrl.1997.v4.n1.a1 1997
[22]

Seeger, T

A. Seeger, T. Tao, and J. Wright,Pointwise convergence of lacunary spherical means, Harmonic anal- ysis at Mount Holyoke (South Hadley, MA, 2001), Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 341–351, DOI 10.1090/conm/320/05617. MR1979950↑3

work page doi:10.1090/conm/320/05617 2001
[23]

E. M. Stein,Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A.73(1976), no. 7, 2174– 2175, DOI 10.1073/pnas.73.7.2174. MR0420116↑1

work page doi:10.1073/pnas.73.7.2174 1976
[24]

E. M. Stein and G. Weiss,Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, vol. No. 32, Princeton University Press, Princeton, NJ, 1971. MR0304972↑8 (R. Basak) DEPARTMENT OFMATHEMATICS, NATIONALTAIWANNORMALUNIVERSITY, NO. 88, SECTION4, TINGZHOUROAD, WENSHANDISTRICT, TAIPEICITY, TAIWAN116, R.O.C. Email address:rijubasak52...

1971