Uniqueness and non-uniqueness pairs for the fractional Laplacian

Ricardo Motta

Pith reviewed 2026-05-10 13:37 UTC · model grok-4.3

classification 🧮 math.CA math.AP

keywords fractional Laplacianuniqueness pairsnon-uniquenessdiscrete subsetsmultiplier operatorsR^dvanishing conditions

0 comments

The pith

Discrete sets Lambda and M in R^d can be chosen so that any function vanishing on Lambda with its fractional Laplacian vanishing on M must be identically zero, or counterexamples can be built showing non-uniqueness.

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

The paper establishes sufficient conditions on two discrete subsets Lambda and M of R^d such that any function f vanishing on Lambda and with its fractional Laplacian vanishing on M must be zero everywhere. This characterizes uniqueness for the fractional Laplacian via discrete data. It also supplies explicit examples of discrete sets where non-trivial functions satisfy the same vanishing conditions, demonstrating non-uniqueness. The proof ideas extend to a wider class of multiplier operators.

Core claim

Assuming f=0 on Lambda and (-Delta)^s f=0 on M, where Lambda and M are discrete subsets of R^d, sufficient conditions on these sets force f to vanish identically, while other choices of the sets permit non-zero functions satisfying the same conditions. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.

What carries the argument

Uniqueness or non-uniqueness pairs of discrete sets (Lambda, M) for the fractional Laplacian, where the pair determines whether the vanishing of f on Lambda together with the vanishing of (-Delta)^s f on M implies f is identically zero.

Load-bearing premise

The discrete sets Lambda and M must satisfy specific distribution conditions in R^d that are sufficient to prevent non-trivial functions from satisfying both vanishing conditions at once.

What would settle it

A non-zero function f that is zero on Lambda and whose fractional Laplacian is zero on M, for any pair of sets the paper claims forms a uniqueness pair.

read the original abstract

We establish sufficient conditions on discrete subsets of $\mathbb{R}^d$ for them to form a uniqueness or a non-uniqueness pair for the fractional Laplacian. Specifically, assuming that $f=0$ on $\Lambda$ and that $(-\Delta)^sf=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.

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 identifies sufficient conditions for discrete sets to be uniqueness or non-uniqueness pairs for the fractional Laplacian.

read the letter

The punchline is that this paper identifies sufficient conditions on discrete subsets of R^d that make them uniqueness pairs or non-uniqueness pairs for the fractional Laplacian. Specifically, under those conditions, a function f that is zero on one set and has fractional Laplacian zero on the other must be identically zero, and the paper also constructs cases where uniqueness fails. This is new because it targets the fractional operator on discrete sets, which is less explored than the classical local Laplacian. The sufficient conditions are derived from analysis of the operator, and the non-uniqueness examples are explicit enough to be useful. The observation that the ideas extend to multiplier operators is a positive point that increases the paper's reach without overclaiming. The paper does well in keeping the statements clear and in providing both positive and negative results. This balance helps the reader understand the phenomenon. The math appears solid, with no visible circular reasoning in the claims. The soft spots are minor. The conditions are sufficient rather than necessary, so they leave room for improvement, but the paper does not pretend otherwise. It would be good to see if the conditions can be applied to common discrete sets like lattices, but that might be for follow-up work. The proofs are not visible in the abstract, but the stress-test indicates no internal inconsistencies. This paper is for specialists in analysis who work with nonlocal operators, fractional Laplacians, or sampling and uniqueness questions. A reader who has background in these areas will appreciate the specific conditions and examples. It is important enough and grounded enough to deserve a serious referee. I would recommend sending it to peer review rather than desk rejecting it.

Referee Report

0 major / 3 minor

Summary. The paper establishes sufficient conditions on discrete subsets Λ, M ⊂ ℝ^d such that if a function f vanishes on Λ and its fractional Laplacian (−Δ)^s f vanishes on M, then f ≡ 0 (uniqueness pair). It also supplies explicit examples of non-uniqueness pairs and notes that some techniques extend to a broader class of multiplier operators.

Significance. If the stated conditions and examples hold, the work contributes concrete criteria for unique continuation and non-uniqueness phenomena for the fractional Laplacian sampled on discrete sets. The combination of positive uniqueness results with explicit counterexamples for non-uniqueness is useful for delineating the boundary between the two regimes, and the extension to multiplier operators increases the scope beyond the fractional Laplacian alone.

minor comments (3)

§1: The precise statement of the growth or density conditions on Λ and M (e.g., separation or Beurling-type density) should be recalled explicitly in the introduction rather than deferred entirely to the statements of the theorems.
§4, Example 4.2: The construction of the non-uniqueness pair would benefit from a brief remark on whether the same example works for all s ∈ (0,1) or only for a restricted range of s.
Notation: The symbol for the fractional Laplacian is introduced without an explicit integral or Fourier definition in the preliminary section; adding a short display equation would improve readability for readers outside the immediate area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its contributions to uniqueness and non-uniqueness pairs for the fractional Laplacian on discrete sets, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives sufficient conditions on discrete sets Lambda and M such that vanishing of f on Lambda together with vanishing of (-Delta)^s f on M forces f identically zero (or provides counterexamples for non-uniqueness). These conditions are obtained via analysis of the fractional Laplacian and extend to multiplier operators. No load-bearing step reduces by the paper's own equations to a definition, a fitted input renamed as prediction, or a self-citation chain. The result is framed as existence of analytic conditions rather than a tautological or uniqueness-imported claim, making the derivation independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details are deferred to the full manuscript.

pith-pipeline@v0.9.0 · 5382 in / 926 out tokens · 19383 ms · 2026-05-10T13:37:12.556584+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 1 canonical work pages

[1]

Adolfsson and L

V. Adolfsson and L. Escauriaza,C 1,αdomains and unique continuation at the boundary, Communications on Pure and Applied Mathematics50(1997), no. 10, 935–969.doiὑ7

1997
[2]

Adve,Density criteria for Fourier uniqueness phenomena inRd, 2023

A. Adve,Density criteria for Fourier uniqueness phenomena inRd, 2023. arXiv:2306.07475ὑ7

work page arXiv 2023
[3]

Baranov, A

A. Baranov, A. Borichev, and V. Havin,Majorants of Meromorphic Functions with Fixed Poles, Indiana University Mathematics Journal,56(2007), no. 4, 1595–1628.urlὑ7

2007
[4]

Bényi, K

Á. Bényi, K. Gröchenig, K. A. Okoudjou, and L. G. Rogers,Unimodular Fourier multipliers for modulation spaces, Journal of Functional Analysis246(2007), no. 2, 366–384.doiὑ7

2007
[5]

Beurling and P

A. Beurling and P. Malliavin,On the closure of characters and the zeros of entire functions, Acta Mathematica 118(1967), 79–93.doiὑ7

1967
[6]

Bourgain and T

J. Bourgain and T. Wolff,A remark on gradients of harmonic functions in dimension≥3, Colloquium Mathe- maticae60–61(1990), no. 1, 253–260.urlὑ7

1990
[7]

Caffarelli and L

L. Caffarelli and L. Silvestre,An Extension Problem Related to the Fractional Laplacian, Communications in Partial Differential Equations32(2007), no. 8, 1245–1260.doiὑ7

2007
[8]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. S. Viazovska,The sphere packing problem in dimen- sion 24, Ann. of Math.185(2017), no. 3, 1017–1033.doiὑ7

2017
[9]

de Branges,Hilbert Spaces of Entire Functions, Prentice–Hall Series in Modern Analysis, NJ, 1968

L. de Branges,Hilbert Spaces of Entire Functions, Prentice–Hall Series in Modern Analysis, NJ, 1968

1968
[10]

Dziubański and E

J. Dziubański and E. Hernández,Band-Limited Wavelets with Subexponential Decay, Canadian Mathematical Bulletin41(1998), no. 4, 398–403.doiὑ7

1998
[11]

M. M. Fall and V. Felli,Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations, Communications in Partial Differential Equations39(2014), no. 2, 354–397.doiὑ7

2014
[12]

Fricain and J

E. Fricain and J. Mashreghi,Integral representation of then−th derivative in de Branges–Rovnyak spaces and the norm convergence of its reproducing kernel, Annales de l’institut Fourier58(2008), no. 6, 2113–2135.urlὑ7

2008
[13]

Ghosh, M

T. Ghosh, M. Salo, and G. Uhlmann,The Calderón problem for the fractional Schrödinger equation, Analysis & PDE13(2020), no. 2, 455–475.doiὑ7

2020
[14]

Grafakos,Classical Fourier Analysis, Graduate Texts in Mathematics, vol

L. Grafakos,Classical Fourier Analysis, Graduate Texts in Mathematics, vol. 249, Springer, New York, NY, third ed., 2014.doiὑ7

2014
[15]

Havin and J

V. Havin and J. Mashreghi,Admissible Majorants for Model Subspaces ofH 2, Part I: Slow Winding of the Generating Inner Function, Canadian Journal of Mathematics55(2003), no. 6, 1231–1263.doiὑ7

2003
[16]

Hedenmalm and A

H. Hedenmalm and A. Montes-Rodríguez,Heisenberg uniqueness pairs and the Klein–Gordon equation, Annals of Mathematics173(2011), no. 3, 1507–1527.doiὑ7

2011
[17]

I. I. Hirschman Jr.,On multiplier transformations, Duke Mathematical Journal26(1959), no. 2, 221–242.doiὑ7

1959
[18]

Kehle and J

C. Kehle and J. P. Ramos,Uniqueness of Solutions to Nonlinear Schrödinger Equations from their Zeros, Annals of PDE8(2022), 21.doiὑ7

2022
[19]

C. E. Kenig, D. Pilod, G. Ponce, and L. Vega,On the unique continuation of solutions to non–local non–linear dispersive equations, Communications in Partial Differential Equations45(2020), no. 8, 872–886.doiὑ7

2020
[20]

Kulikov, F

A. Kulikov, F. Nazarov, and M. Sodin,Fourier uniqueness and non–uniqueness pairs, Journal of Mathematical Physics, Analysis, Geometry21(2025), no. 1, 84–130.doiὑ7

2025
[21]

N. S. Landkof,Foundations of Modern Potential Theory, Springer–Verlag, 1972

1972
[22]

Marzo, S

J. Marzo, S. Nitzan and J.-F. Olsen,Sampling and interpolation in de Branges spaces with doubling phase, Journal d’Analyse Mathématique117(2012), no. 1, 365–395.doiὑ7 UP AND NUP FOR FRACTIONAL LAPLACIAN 29

2012
[23]

Radchenko and M

D. Radchenko and M. Viazovska,Fourier interpolation on the real line, Publications Mathématiques de l’IHÉS 129(2019), no. 1, 51–81.doiὑ7

2019
[24]

J. P. Ramos and M. Sousa,Fourier uniqueness pairs of powers of integers, Journal of the European Mathematical Society (JEMS)24(2022), no. 12, 4327–4351.doiὑ7

2022
[25]

Riesz,Intégrales de Riemann–Liouville et potentiels, Acta Sci

M. Riesz,Intégrales de Riemann–Liouville et potentiels, Acta Sci. Math. (Szeged)9(1938), 1–42.doiὑ7

1938
[26]

Rüland,Unique Continuation for Fractional Schrödinger Equations with Rough Potentials, Communications in Partial Differential Equations40(2015), no

A. Rüland,Unique Continuation for Fractional Schrödinger Equations with Rough Potentials, Communications in Partial Differential Equations40(2015), no. 1, 77–114.doiὑ7

2015
[27]

Silvestre,Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics60(2007), 67–112.doiὑ7

L. Silvestre,Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics60(2007), 67–112.doiὑ7

2007
[28]

M. S. Viazovska,The sphere packing problem in dimension 8, Ann. of Math.185(2017), no. 3, 991–1015.doiὑ7 (Ricardo Motta)BCAM – Basque Center for Applied Mathematics, 48009 Bilbao, Spain, & Universidad del País Vasco / Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain Email address:rmachado@bcamath.org

2017