Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem

David Lee

REVIEW 4 major objections 5 minor 1 cited by

Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper constructs a fundamental solution of the logarithmic Laplacian in every dimension $d\ge 1$ within Lizorkin distributions, and classifies all such solutions as the explicit one plus a surface-layer distribution on the unit sphere.

desk verdict The central distribution in Definition 4 is undefined, so the main existence theorems collapse, but the Liouville argument and the overall strategy are salvageable. read the letter →

arxiv 2506.20121 v1 pith:FKR7H6VT submitted 2025-06-25 math.AP

David Lee This is my paper · ORCID

classification math.AP MSC 35A0835JXX35R1142B3742B1042B20

keywords logarithmicLaplacianfundamentalsolutionLizorkindistributionsdivisionproblemsinglelayerHelmholtzequationLiouvilletheoremnonlocaloperators

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to prove that the logarithmic Laplacian — the operator obtained by differentiating the fractional Laplacian with respect to its exponent at zero — has a fundamental solution in every dimension $d\ge 1$, not just $d\ge 3$ as previously known. The proof recasts the problem as a Fourier division problem $\log(|\xi|^2)\Psi = 1$ and solves it with a renormalized kernel that tames the singularity on the unit sphere. If the argument is right, the set of all fundamental solutions is completely classified: they differ by harmonic functions, and harmonic functions for this operator are exactly generalized eigenfunctions of the ordinary Laplacian with eigenvalue 1. In dimensions 1 and 2 the constructed solution satisfies the decay bound conjectured by Chen and Véron away from the origin, leaving only local integrability near zero open.

What carries the argument

The load-bearing object is the renormalized Fourier kernel $\hat E_{\log}=\hat E_{\log}^1+\hat E_{\log}^2$, defined for Schwartz $\varphi$ by $$\langle \hat E_{\log},\varphi\rangle = \frac12\int_{||\xi|-1|<1}\frac{\varphi(\xi)-\varphi(\xi/|\xi|)}{\log|\xi|}\,d\xi + \frac12\int_{|\xi|>2}\frac{\varphi(\xi)-\varphi(\xi/|\xi|)}{\log|\xi|}\,d\xi.$$ Subtracting the radial projection cancels the pole of $1/\log|\xi|$ on the unit sphere and makes the identity $\log(|\xi|^2)\hat E_{\log}=1$ hold on the Lizorkin test space $\Psi(\mathbb{R}^d)$. The second mechanism is the comparison with the Helmholtz fundamental solution $\Phi$: the difference between the logarithmic and Helmholtz kernels is a single-layer distribution on $S^{d-1}$, i.e. a distribution that acts through a functional on the sphere, and single layers are annihilated by $\log(|\xi|^2)$ on the sphere. That comparison converts the Fourier remainders into Bessel integrals whose asymptotics give the decay in dimensions 1 and 2.

What would settle it

Compute the Definition 4 pairing on $\varphi(\xi)=e^{-|\xi|^2}\tau(\xi/|\xi|)$ with $\tau\in C^\infty(S^{d-1})$ having nonzero spherical mean. The exterior term equals $\frac12\left(\int_{S^{d-1}}\tau\,d\sigma\right)\int_2^\infty \frac{e^{-r^2}-e^{-1}}{\log r}r^{d-1}\,dr$, which diverges to $-\infty$; unless a finite-part prescription is supplied, $\hat E_{\log}$ is not a tempered distribution and the identity $\log(|\xi|^2)\hat E_{\log}=1$ in $\Psi'$ is not established.

Watch

Extended reading notes

Core claim

The central claim is that in the frequency domain the existence of a fundamental solution is a division problem. The paper defines a tempered distribution $\hat E_{\log}$ by pairing a Schwartz function $\varphi$ with $\frac12(\varphi(\xi)-\varphi(\xi/|\xi|))/\log|\xi|$ near the unit sphere and for $|\xi|>2$; subtracting the radial projection removes the singularity at $|\xi|=1$ and makes the Fourier multiplier $\log(|\xi|^2)$ invertible on Lizorkin test functions. It then shows $\log(|\xi|^2)\hat E_{\log}=1$ in $\Psi'(\mathbb{R}^d)$, so $E=\mathcal F^{-1}\hat E_{\log}$ solves $\log(-\Delta)E=\delta_0$ in $Z'(\mathbb{R}^d)$. The Liouville theorem identifies every solution of the homogeneous equation with a single-layer distribution supported on the unit sphere, giving the full classification. For dimensions 1 and 2, subtracting a classical Helmholtz fundamental solution leaves remainder terms whose Fourier inverses are controlled by Bessel estimates, yielding $|E(x)|\le C|x|^{-(d-1)/2}$ for $|x|\ge 2$.

Load-bearing premise

The load-bearing premise is that the pairing written in Definition 4 is a well-defined tempered distribution; for a generic Schwartz test function the exterior integral diverges, so the proof depends on a regularization that the paper does not spell out.

Editorial extensions

If this is right

In every dimension $d\ge 1$, the logarithmic Laplacian has a fundamental solution in the Lizorkin distribution space, extending the previously known range $d\ge 3$.
The full solution set is an affine space over the harmonic functions, and being harmonic for the logarithmic Laplacian is equivalent to being a generalized eigenfunction of $-\Delta$ with eigenvalue 1.
In dimensions 1 and 2, there is a fundamental solution with the conjectured decay $|E(x)|\lesssim |x|^{-(d-1)/2}$ away from the origin, so the Chen–Véron conjecture is reduced to local integrability near zero.
The logarithmic Laplacian is well-defined on $Z'(\mathbb{R}^d)$, even though it cannot act on all tempered distributions because polynomials are not controlled.
The Liouville theorem for the logarithmic Laplacian holds without any boundedness assumption on the solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the classification is correct, selecting a distinguished fundamental solution is the same as selecting a single-layer correction on the unit sphere; a natural next step is to identify the correction that enforces a Sommerfeld-type radiation condition.
Beyond the paper: the same division-problem template — regularize the symbol by subtracting a projection onto the singular set — should apply to other operators whose symbol is non-smooth on a submanifold rather than at a point.
Beyond the paper: the full Chen–Véron conjecture is now pinned to one object, the remainder $G_2$ from Section 3; proving or disproving that $G_2$ is a regular distribution near 0 would settle it.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The stress-test note is right, and the reader's verdict lands in the right place. Definition 4's exterior integral diverges for generic Schwartz test functions because φ(ξ/|ξ|) is constant along rays, and ∫_2^∞ r^{d−1}/log r dr diverges. It also fails for Lizorkin test functions: ψ(ξ)=|ξ|^2 e^{−|ξ|^2} is in Ψ(R^d) but ψ(ξ/|ξ|)=e^{−1}, so the same divergence appears. Thus Ê_log is neither in S'(R^d) nor Ψ'(R^d), and Theorem 2(2) and Theorem 3 do not follow. This is not a minor typo; it is the main object of the paper, and the Section 2 computation is a formal manipulation of something that has not been defined.

That said, the paper is not a waste. The Liouville theorem section is a largely independent argument and reads plausibly: the support of û is confined to the unit sphere (up to a possible point at 0 that the author's convention should explicitly exclude), and the single-layer classification via the Taylor subtraction is a clean idea. That part might survive essentially unchanged once the quotient by polynomials is made precise. The comparison with the Helmholtz fundamental solution in Section 3 is also a good strategy, and it would be a nice way to obtain the d=1,2 bound once a correct regularized inverse of log(|ξ|^2) is available. The author is honest about the unresolved local integrability near zero (Remark 8) and acknowledges an earlier error in the acknowledgments; that is the behavior of someone actually engaging with the mathematics.

Soft spots, in proportion: the divergent exterior integral is load-bearing, and the Section 3 bound estimates an E2_log that differs from the object defined earlier. The support-at-zero omission in the Liouville proof is a smaller gap, fixable by working in the quotient S'/polynomials. The untracked Fourier constants in the bound are a minor inconvenience, unlikely to change the result.

Who is this for? Someone working on the logarithmic Laplacian or on division problems for non-smooth symbols might find the strategy and the Liouville part useful as a starting point, but the paper in its current form should not be the cited reference for the main existence theorem. I would not cite it yet. I would, however, send it to a serious referee: the flaw is technical and possibly repairable, and the Liouville argument deserves scrutiny and probably publication after revision. My recommendation is reject in current form, but with a clear and constructive path toward a fix.

Referee Report

4 major / 5 minor

Summary. The paper proposes an alternative construction of a fundamental solution for the logarithmic Laplacian in all dimensions by solving the division problem log(|·|^2)Ê = 1 in the space of Lizorkin distributions. It defines a candidate distribution Ê_log via two subtracted integrals, claims that this object yields a fundamental solution, proves a Liouville theorem identifying harmonic distributions with single layers on the unit sphere, and uses these tools to give a partial resolution of the Chen–Véron conjecture in dimensions 1 and 2. The approach is inspired by the Malgrange–Ehrenpreis theory and is developed through a comparison with the classical Helmholtz fundamental solution.

Significance. If correct, the paper would extend the existence of fundamental solutions of the logarithmic Laplacian to all dimensions, provide a new Fourier-side proof of a Liouville theorem, and make progress on the Chen–Véron conjecture in dimensions 1 and 2. The strategy of renormalizing the inverse logarithmic symbol and comparing with the Helmholtz solution is a natural and potentially valuable idea. However, the central object of the paper, Ê_log, is not a well-defined distribution, so the main theorems are not established as written. The paper does not include machine-checked proofs or reproducible code; its contribution is purely analytic and, in its current form, the core construction fails.

major comments (4)

[Definition 4] The pairing ⟨Ê^2_log, φ⟩ = (1/2)∫_{|ξ|>2} [φ(ξ) − φ(ξ/|ξ|)]/log|ξ| dξ is divergent for any φ ∈ S(R^d) whose restriction to the unit sphere is not identically zero, because φ(ξ/|ξ|) is independent of |ξ| and ∫_2^∞ r^{d−1}/log r dr diverges for every d ≥ 1. Such φ also exist in the Lizorkin space Ψ(R^d), for instance a smooth function supported in an annulus containing S^{d−1} with nonvanishing values on the sphere. Hence Ê_log is not a distribution on S′(R^d) or Ψ′(R^d), and the existence step in Section 2 applies log(|·|^2) to an undefined object.
[Section 2, proof of existence] The computation of ⟨log(|·|^2)Ê_log, ψ⟩ discards the subtraction terms ψ(ξ/|ξ|). After multiplication by log(|ξ|^2), these terms become 2ψ(ξ/|ξ|), and the exterior contribution ∫_{|ξ|>2} 2ψ(ξ/|ξ|) dξ is not zero and, as written, diverges for generic ψ. The claimed identity ⟨log(|·|^2)Ê_log, ψ⟩ = ∫ψ dξ is therefore not a consequence of Definition 4 and is not justified by the displayed calculation.
[Section 3, proof of Theorem 3] The object called E^2_log in the proof of Theorem 3 is the inverse Fourier transform of the locally integrable function 1_{|ξ|>2}/log(|ξ|^2), which is different from the Ê^2_log of Definition 4, where the subtraction φ(ξ/|ξ|) appears. The bounds established for this different object do not control the fundamental solution constructed from Ê_log, and the decomposition E = Φ + E^1_rem + E^2_rem does not follow from Theorem 2(2) without the ill-defined Ê_log.
[Section 2, proof of the Liouville theorem, Step 2] The identity ψ − Eτ = (1−Λ)ψ + ψ̃ and the assertion ⟨û, (1−Λ)ψ⟩ = 0 from supp(û) ⊂ S^{d−1} require that (1−Λ)ψ vanish in a neighborhood of S^{d−1}. The definition of Λ in Definition 5 only ensures Λ = 1 on S^{d−1}, not in a neighborhood; a distribution supported on S^{d−1} can pair nontrivially with a function that merely vanishes on the surface, as the radial derivative example in Remark 6 illustrates. This gap is repairable by requiring Λ ≡ 1 near S^{d−1}, but the proof as written is incomplete.

minor comments (5)

[Section 2] The notation Ê_Flog appears where Ê_log is meant; please correct the typo.
[Section 1.3] The outline refers to 'Appendix 3', but the appendix is unnumbered; renumber the cross-reference.
[Definition 4] The region ||ξ|−1| < 1 is the annulus 0 < |ξ| < 2; this should be stated explicitly, since the origin is not included but lies in the closure.
[Section 3] The phrase 'the unique extension from Z′(R^d) to S′(R^d)' in the paragraph after (12) needs a precise statement; the convention in Definition 3 and Remark 3 does not define a canonical extension for all elements of Z′(R^d).
[Definition 5] The citation '[1, Chapter 6]' does not match the bibliography format; assign Strichartz a proper reference key and use it consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is a self-contained attempt at a first-principles derivation; its central gaps are mathematical errors, not circular reductions.

full rationale

The paper contains no fitted parameters that are renamed as predictions, no self-citation chain that carries the argument, and no normalization chosen to force the advertised conclusion. The proposed fundamental solution in Definition 4 is introduced explicitly as a guess and then tested directly: the proof of existence computes log(|cdot|^2) times the candidate and verifies that it evaluates to 1 on Lizorkin test functions. The Liouville theorem and the classification of fundamental solutions in Theorem 2 are proved in the paper itself by a support argument followed by a single-layer representation; they are not imported from prior work by the author. The Helmholtz analogue is cited to Schwartz as external mathematical background, and the comparison in Theorem 3 between the logarithmic and Helmholtz fundamental solutions is a derivation, not an assumption. The main defect identified by a rigorous reading is that the integrals in Definition 4 do not converge for generic Schwartz or even Lizorkin test functions, so the object E_log may not be a distribution and the computation in Section 2 may act on an undefined object. That is a correctness gap or error in the proof as written, not a circular reduction: the conclusion is not assumed in the construction, and no input is fitted to the desired output. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters and no fitted constants. The central claim rests on the ad hoc regularization hat E_log, which is not a valid distribution as written, and on standard Fourier/distribution background. No new physical entities are introduced.

assumptions (5)

standard math Fourier transform duality and the Lizorkin space correspondence (Z maps to Psi)
Used in Section 2 to translate log(-Delta) E = delta_0 into log(|xi|^2) hat E = 1 in Psi'.
standard math Malgrange-Ehrenpreis theorem and the classical Helmholtz fundamental solution classification from Schwartz [Sch66]
Used in Section 3 to select Phi and compare spectra of the logarithmic Laplacian and the Helmholtz operator.
domain assumption Unique representative convention: from [u] in Z' choose the representative whose Fourier transform is locally integrable near 0
Definition 3 and Remark 3 exclude polynomials from harmonic functions and allow extension from Z' to S'.
ad hoc to paper The renormalized inverse hat E_log defined by the subtracted integrals is a tempered distribution satisfying the division identity
Definition 4 and the direct computation in Section 2. This load-bearing premise is false as written because the tail integral diverges.
standard math Bessel function asymptotics |J0(t)| <= C t^{-1/2} and Hankel function behavior
Used in Section 3 Step 2 to estimate E2_Helm and E2_log.

invented entities (1)

The renormalized inverse distribution hat E_log
purpose: Solve log(|xi|^2) hat E = 1 in Psi' so its inverse Fourier transform is a fundamental solution.
Definition 4 introduces it as a finite sum of angular-subtracted integrals. As written the |xi|>2 term is not a tempered distribution; no external or machine-checked evidence supports its existence.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem." pith.science (2026). https://pith.science/paper/FKR7H6VT

@misc{pith2026250620121,
  author       = {Pith},
  title        = {Pith review of: Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKR7H6VT}},
  note         = {Machine review of arXiv:2506.20121}
}

abstract

Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \geq 3$) was established by Huyuan Chen and Laurent V\'eron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and V\'eron regarding the behavior of solutions in dimensions 1 and 2.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Extension theorems for logarithmic Schr\"odinger and discrete Laplacian operators
math.CA 2026-04 unverdicted novelty 6.0 of 10

Logarithmic operators log L_V and log(−Δ_d) are realized as boundary values of solutions to suitable extension problems, in a more involved way than the fractional Laplacian case.

Reference graph

Works this paper leans on

20 extracted references · 9 canonical work pages · cited by 1 Pith paper

[1]

A guide to distribution theory and Fourier transforms

Strichartz, R. A guide to distribution theory and Fourier transforms. (World Scientific Publishing Co., Inc., River Edge, NJ,2003), https://doi.org/10.1142/5314, Reprint of the 1994 original [CRC, Boca Raton; MR1276724 (95f:42001)]

doi:10.1142/5314 2003
[2]

Representation theorems for solutions of the H elmholtz equation on R ^n

Shmuel Agmon. Representation theorems for solutions of the H elmholtz equation on R ^n . In Differential operators and spectral theory , volume 189 of Amer. Math. Soc. Transl. Ser. 2 , pages 27--43. Amer. Math. Soc., Providence, RI, 1999

work page 1999
[3]

Helmholtz equations for the L aplace operator and its powers

Mohamed Ben Chrouda. Helmholtz equations for the L aplace operator and its powers. Mathematica , 66(89)(2):210--215, 2024

2024
[4]

Localization for general H elmholtz

Xinyu Cheng, Dong Li, and Wen Yang. Localization for general H elmholtz. J. Differential Equations , 393:139--154, 2024

2024
[5]

The C auchy problem associated to the logarithmic L aplacian with an application to the fundamental solution

Huyuan Chen and Laurent V\' e ron. The C auchy problem associated to the logarithmic L aplacian with an application to the fundamental solution. J. Funct. Anal. , 287(3):Paper No. 110470, 72, 2024

work page 2024
[6]

The D irichlet problem for the logarithmic L aplacian

Huyuan Chen and Tobias Weth. The D irichlet problem for the logarithmic L aplacian. Comm. Partial Differential Equations , 44(11):1100--1139, 2019

2019
[7]

Solution of some problems of division

Leon Ehrenpreis. Solution of some problems of division. I . D ivision by a polynomial of derivation. Amer. J. Math. , 76:883--903, 1954

work page 1954
[8]

Solution of some problems of division

Leon Ehrenpreis. Solution of some problems of division. II . D ivision by a punctual distribution. Amer. J. Math. , 77, 1955

work page 1955

Show all 20 references

[9]

Entire s -harmonic functions are affine

Mouhamed Moustapha Fall. Entire s -harmonic functions are affine. Proc. Amer. Math. Soc. , 144(6):2587--2592, 2016

2016
[10]

Liouville theorems for a general class of nonlocal operators

Mouhamed Moustapha Fall and Tobias Weth. Liouville theorems for a general class of nonlocal operators. Potential Anal. , 45(1):187--200, 2016

2016
[11]

Helmholtz solutions for the fractional L aplacian and other related operators

Vincent Guan, Mathav Murugan, and Juncheng Wei. Helmholtz solutions for the fractional L aplacian and other related operators. Commun. Contemp. Math. , 25(2):Paper No. 2250016, 18, 2023

2023
[12]

Classical F ourier analysis , volume 249 of Graduate Texts in Mathematics

Loukas Grafakos. Classical F ourier analysis , volume 249 of Graduate Texts in Mathematics . Springer, New York, third edition, 2014

2014
[13]

Characterization of solutions of a generalized helmholtz problem

Daniel Hauer. Characterization of solutions of a generalized helmholtz problem. arXiv preprint arXiv:2306.17562 , 2023

2023 arXiv
[14]

Existence et approximation des solutions des \' e quations aux d\' e riv\' e es partielles et des \' e quations de convolution

Bernard Malgrange. Existence et approximation des solutions des \' e quations aux d\' e riv\' e es partielles et des \' e quations de convolution. Ann. Inst. Fourier (Grenoble) , 6:271--355, 1955/56

1955
[15]

Distributions, partial differential equations, and harmonic analysis

Dorina Mitrea. Distributions, partial differential equations, and harmonic analysis . Universitext. Springer, Cham, 2018. Second edition [ MR3114783]

2018
[16]

Fundamental solutions of linear partial differential operators

Norbert Ortner and Peter Wagner. Fundamental solutions of linear partial differential operators . Springer, Cham, 2015. Theory and practice

2015
[17]

Regularised integrals, sums and traces , volume 59 of University Lecture Series

Sylvie Paycha. Regularised integrals, sums and traces , volume 59 of University Lecture Series . American Mathematical Society, Providence, RI, 2012. An analytic point of view

2012
[18]

Stefan G. Samko. Hypersingular integrals and their applications , volume 5 of Analytical Methods and Special Functions . Taylor & Francis Group, London, 2002

2002
[19]

Th\' e orie des distributions

Laurent Schwartz. Th\' e orie des distributions . Publications de l'Institut de Math\' e matique de l'Universit\' e de Strasbourg, IX-X. Hermann, Paris, 1966. Nouvelle \' e dition, enti\' e rement corrig\' e e, refondue et augment\' e e

1966
[20]

Convergence rates of the fractional to the local dirichlet problem, 2024

Leon Bungert and Félix del Teso. Convergence rates of the fractional to the local dirichlet problem, 2024

2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

A guide to distribution theory and Fourier transforms

Strichartz, R. A guide to distribution theory and Fourier transforms. (World Scientific Publishing Co., Inc., River Edge, NJ,2003), https://doi.org/10.1142/5314, Reprint of the 1994 original [CRC, Boca Raton; MR1276724 (95f:42001)]

doi:10.1142/5314 2003

[2] [2]

Representation theorems for solutions of the H elmholtz equation on R ^n

Shmuel Agmon. Representation theorems for solutions of the H elmholtz equation on R ^n . In Differential operators and spectral theory , volume 189 of Amer. Math. Soc. Transl. Ser. 2 , pages 27--43. Amer. Math. Soc., Providence, RI, 1999

work page 1999

[3] [3]

Helmholtz equations for the L aplace operator and its powers

Mohamed Ben Chrouda. Helmholtz equations for the L aplace operator and its powers. Mathematica , 66(89)(2):210--215, 2024

2024

[4] [4]

Localization for general H elmholtz

Xinyu Cheng, Dong Li, and Wen Yang. Localization for general H elmholtz. J. Differential Equations , 393:139--154, 2024

2024

[5] [5]

The C auchy problem associated to the logarithmic L aplacian with an application to the fundamental solution

Huyuan Chen and Laurent V\' e ron. The C auchy problem associated to the logarithmic L aplacian with an application to the fundamental solution. J. Funct. Anal. , 287(3):Paper No. 110470, 72, 2024

work page 2024

[6] [6]

The D irichlet problem for the logarithmic L aplacian

Huyuan Chen and Tobias Weth. The D irichlet problem for the logarithmic L aplacian. Comm. Partial Differential Equations , 44(11):1100--1139, 2019

2019

[7] [7]

Solution of some problems of division

Leon Ehrenpreis. Solution of some problems of division. I . D ivision by a polynomial of derivation. Amer. J. Math. , 76:883--903, 1954

work page 1954

[8] [8]

Solution of some problems of division

Leon Ehrenpreis. Solution of some problems of division. II . D ivision by a punctual distribution. Amer. J. Math. , 77, 1955

work page 1955