arxiv: 2605.09809 · v1 · submitted 2026-05-10 · 🧮 math.CA

Recognition: 2 theorem links

· Lean Theorem

Sharpness of convolution bounds for measures

Sanghyuk Lee, Sungchul Lee

Pith reviewed 2026-05-12 02:45 UTC · model grok-4.3

classification 🧮 math.CA

keywords convolution operatorsfractal measuresFrostman conditionFourier decayL^p L^q boundsrestriction estimatesharmonic analysis

0 comments

The pith

The sharp (p,q) range is determined for L^p to L^q bounds on convolution with fractal measures satisfying α-Frostman and β/2-Fourier decay conditions.

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

The paper determines the precise values of p and q for which the operator sending a function f to its convolution with such a measure μ is bounded from L^p to L^q. These measures are compactly supported probability measures on R^d that obey both a mass distribution bound of order α and a decay bound on their Fourier transform of order β/2. A reader might care because convolution bounds with measures of this type control many questions about how functions interact with sets of fractional dimension. The result is proved by constructing explicit examples that meet the conditions and a lower regularity requirement, which also sharpens earlier work on restriction estimates.

Core claim

We determine the sharp (p,q) range for L^p--L^q bounds of convolution operators f↦μ*f associated with fractal measures μ∈P_{α,β}(R^d), namely, compactly supported Borel probability measures satisfying the α-Frostman condition μ(B(x,ρ)) ≲ ρ^α and the β/2-Fourier decay condition |μ̂(ξ)| ≲ |ξ|^{-β/2}. Sharpness is established by constructing measures satisfying these conditions together with a suitable lower regularity condition. Modifications of the same constructions also refine previous sharpness results for the L^2 restriction estimate of Mockenhaupt--Mitsis--Bak--Seeger by producing, in every dimension and in both the geometric (α≥β) and non-geometric (β>α) regimes, a single measure in P_{

What carries the argument

The class P_{α,β}(R^d) of measures obeying the α-Frostman condition and the β/2-Fourier decay condition, with constructions that additionally satisfy a lower regularity condition to achieve sharpness.

If this is right

The (p,q) range is sharp for all such measures in both geometric (α ≥ β) and non-geometric (β > α) cases.
Explicit constructions provide a single measure that makes the L^2 restriction threshold sharp in every dimension.
The same constructions work uniformly across dimensions d.
Previous sharpness results for restriction estimates are refined by using one measure instead of possibly different ones.

Where Pith is reading between the lines

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

The constructions might be adaptable to prove sharpness for related operators such as maximal averages or singular integrals.
Similar techniques could apply to measures with different types of decay conditions beyond Fourier.
Testing the bounds with specific self-similar measures like iterated function systems could provide numerical evidence for the thresholds.

Load-bearing premise

Sharpness holds because there exist measures in the class P_{α,β} that also obey a suitable lower regularity condition allowing the lower bounds to match the upper bounds.

What would settle it

A direct computation showing that for one of the constructed measures the convolution operator fails to be bounded at an exponent on the claimed boundary would falsify the sharpness result.

Figures

Figures reproduced from arXiv: 2605.09809 by Sanghyuk Lee, Sungchul Lee.

**Figure 1.** Figure 1: ∆α,β: the range of p, q in Theorem 1.3 Convolution with general measure. The proof of the sufficiency naturally leads to a generalization obtained by considering compactly supported Borel probability measures µ on R d satisfying the following two quantitative conditions: • α-Frostman condition: (1.1) µ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: The geometric case Geometric case. Our first result is the following, which shows the range of p, q in Theorem 1.3 is optimal for certain measures. Theorem 1.4. Suppose that 0 < β ≤ α < d. Then there exists µ ∈ Pα,β(R d ) with dimH(supp µ) = α such that (1.4) holds if and only if (1/p, 1/q) ∈ ∆α,β. Since every measure in Pα,β(R d ) is α-Frostman, its support must have Hausdorff dimension at least α. Thus, … view at source ↗

**Figure 3.** Figure 3: Nongeometric case Proposition 1.8 provides a measure µ ∈ Pα,β(R d ) whose ambient support has Hausdorff dimension β, but which still contains a near s-AD regular heavy core F on which µ carries nearly α-dimensional mass at every small scale. This heavy core is the key to proving sharpness in the nongeometric case: when used as a test set, it yields the sharp necessary condition for the convolution bound be… view at source ↗

read the original abstract

In this paper, we determine the sharp \((p,q)\) range for \(L^p\)--\(L^q\) bounds of convolution operators \(f\mapsto \mu*f\) associated with fractal measures \(\mu\in \mathcal P_{\alpha,\beta}(\mathbb R^d)\), namely, compactly supported Borel probability measures satisfying the \(\alpha\)-Frostman condition \[ \mu(B(x,\rho)) \lesssim \rho^\alpha, \qquad \forall (x,\rho)\in \mathbb R^d\times (0,1), \] and the \(\beta/2\)-Fourier decay condition \[ |\widehat{\mu}(\xi)| \lesssim |\xi|^{-\beta/2}, \qquad \forall \xi\in\mathbb R^d. \] Sharpness is established by constructing measures satisfying these conditions together with a suitable lower regularity condition. Modifications of the same constructions also refine previous sharpness results for the \(L^2\) restriction estimate of Mockenhaupt--Mitsis--Bak--Seeger by producing, in every dimension and in both the geometric \((\alpha\ge\beta)\) and non-geometric \((\beta>\alpha)\) regimes, a single measure in \(\mathcal P_{\alpha,\beta}(\mathbb R^d)\) for which the corresponding threshold exponent is sharp.

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 settles the sharp (p,q) range for these convolution bounds by giving explicit constructions that match the upper bounds from the Frostman and Fourier hypotheses in every dimension and both regimes.

read the letter

The main takeaway is that they pin down exactly when μ * f stays bounded from L^p to L^q for measures that obey both the α-Frostman condition and the β/2 Fourier decay. The upper bound follows from the usual Fourier multiplier estimates once those two hypotheses are in place. For sharpness they produce concrete examples in P_{α,β} that also satisfy an extra lower-regularity condition, and these examples work uniformly in all dimensions, whether α ≥ β or β > α. The same families tighten the known sharpness examples for the L^2 restriction theorem of Mockenhaupt-Mitsis-Bak-Seeger, again with a single construction covering both regimes. That unification is the clearest advance over the earlier literature. The positive direction is standard once the hypotheses are granted, and the lower bounds rest on explicit constructions rather than abstract existence arguments, which makes the claim easier to check. A possible soft spot is how much the added lower-regularity condition narrows the parameter range in practice, but the constructions are described as being carried out directly without visible gaps between the upper and lower bounds. This is for people working on restriction problems and fractal measures in harmonic analysis. The claim is precise, the method is direct, and the constructions are the kind of thing a referee can verify in detail, so it deserves peer review.

Referee Report

0 major / 3 minor

Summary. The paper determines the sharp (p,q) range for the L^p to L^q boundedness of the convolution operator f ↦ μ * f associated to measures μ in the class P_{α,β}(R^d) of compactly supported probability measures satisfying the α-Frostman condition μ(B(x,ρ)) ≲ ρ^α and the β/2-Fourier decay |μ̂(ξ)| ≲ |ξ|^{-β/2}. The positive direction follows from these two hypotheses by standard harmonic-analysis arguments; sharpness is obtained by explicit constructions of measures obeying the same upper conditions together with an additional lower-regularity condition. The constructions are given in every dimension and separately in the geometric regime α ≥ β and the non-geometric regime β > α. The same examples also sharpen the L^2 restriction theorem of Mockenhaupt–Mitsis–Bak–Seeger in both regimes.

Significance. If the constructions are correct, the work supplies a complete, parameter-sharp characterization of the convolution bounds under the stated Frostman and Fourier-decay hypotheses. The explicit, dimension-independent constructions that simultaneously achieve sharpness for both the convolution and the restriction problems constitute a concrete advance over previous partial results.

minor comments (3)

The lower-regularity condition required for the sharpness constructions is invoked repeatedly but is never stated as a numbered definition or displayed equation; inserting a clear, self-contained statement (e.g., as Definition 2.4) would improve readability.
In the abstract and introduction the class is denoted P_{α,β}(R^d) without an immediate reference to the precise definition; a parenthetical reminder of the two defining inequalities would help readers who consult only the abstract.
The paper distinguishes the geometric and non-geometric regimes but does not include a short table or diagram summarizing the resulting (p,q) thresholds in each regime; such a summary would make the main theorem easier to parse.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard tools and explicit constructions

full rationale

The paper derives the positive L^p-L^q bounds from the α-Frostman and β/2-Fourier decay hypotheses using standard harmonic-analysis estimates, then establishes sharpness independently by constructing explicit measures in P_{α,β}(R^d) obeying an additional lower-regularity condition. These constructions are given separately for geometric (α ≥ β) and non-geometric (β > α) cases in every dimension and do not reduce to the upper-bound results, fitted parameters, or self-citations. The central claim therefore rests on externally verifiable constructions rather than any definitional loop or imported uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard tools of harmonic analysis without introducing new free parameters or invented entities.

axioms (1)

standard math Fourier transform and convolution are well-defined and continuous for compactly supported Borel probability measures on R^d
Implicit throughout the statement of bounds and decay conditions.

pith-pipeline@v0.9.0 · 5522 in / 1104 out tokens · 98701 ms · 2026-05-12T02:45:13.836545+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Bak and A

J.-G. Bak and A. Seeger, Extensions of the Stein–Tomas theorem,Math. Res. Lett.18 (2011), no. 4, 767–781

work page 2011
[2]

Bluhm, Random recursive construction of Salem sets,Ark

C. Bluhm, Random recursive construction of Salem sets,Ark. Mat.34 (1996), no. 1, 51–63

work page 1996
[3]

Boucheron, G

S. Boucheron, G. Lugosi, and P. Massart,Concentration Inequalities: A Nonasymptotic The- ory of Independence, Oxford Univ. Press, Oxford, 2013

work page 2013
[4]

Carnovale, J

M. Carnovale, J. M. Fraser, and A. E. de Orellana, Obtaining the Fourier spectrum via Fourier coefficients,Proc. Amer. Math. Soc.(2026), Published electronically March 10, DOI: 10.1090/proc/17610

work page doi:10.1090/proc/17610 2026
[5]

Chen, A Fourier restriction theorem based on convolution powers,Proc

X. Chen, A Fourier restriction theorem based on convolution powers,Proc. Amer. Math. Soc. 142 (2014), no. 11, 3897–3901

work page 2014
[6]

Chen, Sets of Salem type and sharpness of theL 2-Fourier restriction theorem,Trans

X. Chen, Sets of Salem type and sharpness of theL 2-Fourier restriction theorem,Trans. Amer. Math. Soc.368 (2016), no. 3, 1959–1977

work page 2016
[7]

Chen and A

X. Chen and A. Seeger, Convolution Powers of Salem Measures With Applications,Canad. J. Math.69 (2017), no. 2, 284–320

work page 2017
[8]

Christ, A convolution inequality concerning Cantor-Lebesgue measures,Rev

M. Christ, A convolution inequality concerning Cantor-Lebesgue measures,Rev. Mat. Iberoam.1 (1985), no. 4, 79–83

work page 1985
[9]

J. M. Fraser, The Fourier spectrum and sumset type problems,Math. Ann.390 (2024), no. 3, 3891–3930

work page 2024
[10]

Fraser, K

R. Fraser, K. Hambrook, and D. Ryou, Fourier restriction and well-approximable numbers, Math. Ann.391 (2025), no. 3, 4233–4269. CONVOLUTION BOUND 61

work page 2025
[11]

Fraser, K

R. Fraser, K. Hambrook, and D. Ryou, Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger Fourier restriction theorem in all dimensions, arXiv:2505.19526, 2025

work page arXiv 2025
[12]

Hambrook,Restriction theorems and Salem sets, Ph.D

K. Hambrook,Restriction theorems and Salem sets, Ph.D. thesis, University of British Columbia, 2015

work page 2015
[13]

Hambrook and I

K. Hambrook and I. Laba, On the Sharpness of Mockenhaupt’s Restriction Theorem,Geom. Funct. Anal.23 (2013), no. 4, 1262–1277

work page 2013
[14]

Hambrook and I

K. Hambrook and I. Laba, Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions,Bull. Lond. Math. Soc.48 (2016), no. 5, 757–770

work page 2016
[15]

He, Orthogonal projections of discretized sets,J

W. He, Orthogonal projections of discretized sets,J. Fractal Geom.7 (2020), no. 3, 271–317

work page 2020
[16]

B. A. Kpata, I. Fofana, and K. Koua, Necessary condition for measures which are (L q, Lp) multipliers,Ann. Math. Blaise Pascal16 (2009), no. 2, 339–353

work page 2009
[17]

K¨ aenm¨ aki and J

A. K¨ aenm¨ aki and J. Lehrb¨ ack, Measures with predetermined regularity and inhomogeneous self-similar sets,Ark. Mat.55 (2017), no. 1, 165–184

work page 2017
[18]

K¨ aenm¨ aki, J

A. K¨ aenm¨ aki, J. Lehrb¨ ack, and M. Vuorinen, Dimensions, Whitney covers, and tubular neigh- borhoods,Indiana Univ. Math. J.62 (2013), no. 6, 1861–1889

work page 2013
[19]

Laba and M

I. Laba and M. Pramanik, Arithmetic progressions in sets of fractional dimension,Geom. Funct. Anal.19 (2009), no. 2, 429–456

work page 2009
[20]

Li and B

L. Li and B. Liu, Dimension of diophantine approximation and applications, arXiv:2409.12826, 2024

work page arXiv 2024
[21]

Littman,L p–Lq-estimates for singular integral operators arising from hyperbolic equa- tions, inPartial differential equations, Proc

W. Littman,L p–Lq-estimates for singular integral operators arising from hyperbolic equa- tions, inPartial differential equations, Proc. Sympos. Pure Math., Vol. XXIII, Amer. Math. Soc., Providence, RI, 1973, pp. 479–481. Proc. Sympos. Pure Math., Vol. XXIII, Univ. Cali- fornia, Berkeley, Calif., 1971

work page 1973
[22]

Mattila,Fourier Analysis and Hausdorff Dimension, Cambridge Stud

P. Mattila,Fourier Analysis and Hausdorff Dimension, Cambridge Stud. Adv. Math., vol. 150, Cambridge Univ. Press, Cambridge, 2015

work page 2015
[23]

Migliorati, F

G. Migliorati, F. Nobile, and R. Tempone, Convergence estimates in probability and in ex- pectation for discrete least squares with noisy evaluations at random points,J. Multivariate Anal.142 (2015), 167–182

work page 2015
[24]

Mitsis, A Stein-Tomas restriction theorem for general measures,Publ

T. Mitsis, A Stein-Tomas restriction theorem for general measures,Publ. Math. Debrecen60 (2002), no. 1–2, 89–99

work page 2002
[25]

Mockenhaupt, Salem sets and restriction properties of Fourier transforms,Geom

G. Mockenhaupt, Salem sets and restriction properties of Fourier transforms,Geom. Funct. Anal.10 (2000), no. 6, 1579–1587

work page 2000
[26]

Oberlin, A convolution property of the Cantor-Lebesgue measure,Colloq

D. Oberlin, A convolution property of the Cantor-Lebesgue measure,Colloq. Math.47 (1982), no. 1, 113–117

work page 1982
[27]

Oberlin, Convolution estimates for some measures on curves,Proc

D. Oberlin, Convolution estimates for some measures on curves,Proc. Amer. Math. Soc.99 (1987), no. 1, 56–60

work page 1987
[28]

Oberlin, Affine dimension: measuring the vestiges of curvature,Michigan Math

D. Oberlin, Affine dimension: measuring the vestiges of curvature,Michigan Math. J.51 (2003), no. 1, 13–26

work page 2003
[29]

Schrijver,A Course in Combinatorial Optimization, Lecture notes, CWI and University of Amsterdam, Amsterdam, 2017, pp

A. Schrijver,A Course in Combinatorial Optimization, Lecture notes, CWI and University of Amsterdam, Amsterdam, 2017, pp. 134–144

work page 2017
[30]

Shmerkin and V

P. Shmerkin and V. Suomala, A class of random Cantor measures, with applications, in Recent Developments in Fractals and Related Fields, Trends Math., Birkh¨ auser/Springer, Cham, 2017, pp. 233–260

work page 2017
[31]

Shmerkin and V

P. Shmerkin and V. Suomala, Spatially independent martingales, intersections, and applica- tions,Mem. Amer. Math. Soc.251 (2018), no. 1195

work page 2018
[32]

Strichartz, Convolutions with kernels having singularities on a sphere,Trans

R. Strichartz, Convolutions with kernels having singularities on a sphere,Trans. Amer. Math. Soc.148 (1970), 461–471

work page 1970
[33]

T. H. Wolff,Lectures on Harmonic Analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. (Sanghyuk Lee)Department of Mathematical Sciences and RIM, Seoul National Uni- versity, Seoul 08826, Republic of Korea Email address:shklee@snu.ac.kr (Sungchul Lee)Department of Mathematical Sciences, Seoul National Universit...

work page 2003