The Law of the iterated logaritm for smooth functions

Artur Nicolau; Jos\'e G. Llorente

arxiv: 2605.20018 · v1 · pith:K7YHGHAFnew · submitted 2026-05-19 · 🧮 math.CA

The Law of the iterated logaritm for smooth functions

Jos\'e G. Llorente , Artur Nicolau This is my paper

Pith reviewed 2026-05-20 03:55 UTC · model grok-4.3

classification 🧮 math.CA

keywords law of the iterated logarithmsmooth functionsupper half-spacegradient conditionsself-improvementnon-harmonic functionsreal analysis

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The pith

Smooth functions in the upper half-space satisfy a version of the law of the iterated logarithm when their gradients meet certain size conditions.

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

The paper establishes a version of the law of the iterated logarithm that holds for smooth functions in the upper half-space. This matters because it extends the result beyond harmonic functions to more general smooth functions that obey bounds on the size of their gradients and the gradients of their Laplacians. A sympathetic reader would care because these conditions lead to self-improvement in the growth properties of the functions. The result is then applied to situations without harmonicity.

Core claim

The authors prove a version of the Law of the Iterated Logarithm for smooth functions in the upper-half space. As a consequence, certain size conditions on the gradient and the gradient of the Laplacian of a smooth function lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.

What carries the argument

A version of the law of the iterated logarithm adapted to smooth functions in the upper half-space via size conditions on the gradient and the gradient of the Laplacian.

If this is right

Size conditions on the gradient and the gradient of the Laplacian imply self-improving growth properties for the smooth function.
The law of the iterated logarithm conclusion holds in settings without harmonicity.
The results open applications to broader classes of smooth functions in the upper half-space.

Where Pith is reading between the lines

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

The same gradient conditions might produce analogous iterated-logarithm bounds in other domains or for functions with reduced regularity.
This extension could link to boundary-value problems where harmonicity fails but controlled growth still occurs.

Load-bearing premise

The smooth functions in the upper-half space satisfy the stated size conditions on the gradient and the gradient of the Laplacian.

What would settle it

A smooth function in the upper half-space that meets the gradient size conditions but fails to obey the iterated logarithm growth bound on its values or oscillations would disprove the claim.

read the original abstract

A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.

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 extends the LIL to smooth non-harmonic functions in the half-space and derives self-improvement from gradient bounds, but the absorption of Laplacian errors needs checking.

read the letter

The main point is that the authors prove a version of the law of the iterated logarithm for smooth functions in the upper half-space without assuming harmonicity. They then show that size conditions on the gradient and the gradient of the Laplacian produce self-improvement of growth, and they apply the result in settings where harmonicity is absent. This is the concrete advance. Most prior LIL statements in this geometry rely on the mean-value property and maximum principles that come with Δu = 0. Dropping that assumption widens the scope to general smooth functions, which matters for some boundary-value or PDE questions. The self-improvement step is presented as a direct consequence, and that framing is straightforward. The soft spot is whether the stated bounds on |∇u| and |∇Δu| are enough to close the iteration when the Laplacian term is present. Without harmonicity there is no automatic cancellation or averaging, so the extra error must be absorbed uniformly near the boundary or at infinity. If that control is only lower-order and not uniform, the precise limsup = 1 behavior can fail to hold. The abstract gives no explicit indication of how this is handled, so the argument rests on the details of the proof. No circularity or invented quantities appear in the claim. The work is aimed at analysts who track growth estimates and boundary behavior for non-harmonic functions. A reader who already knows the classical harmonic case would see the value in checking whether the extension really works. I would send the paper to peer review. The claim is specific, the setting is standard, and referees can test the key step on the non-harmonic error term.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a version of the Law of the Iterated Logarithm for smooth (not necessarily harmonic) functions in the upper half-space. It further claims that pointwise or integral size conditions on |∇u| and |∇Δu| imply self-improvement of growth, yielding the precise limsup behavior, and applies the result in settings where harmonicity is absent.

Significance. If the central derivation holds, the result would extend deterministic LIL statements beyond the harmonic case, offering a tool for controlling growth of smooth functions via derivative bounds alone. This could be useful in non-harmonic PDE contexts or approximation problems where maximum principles are unavailable.

major comments (2)

[Main proof (Section 3)] The argument that the size bounds on |∇u| and |∇Δu| close the LIL iteration for non-harmonic functions lacks explicit absorption estimates for the ∇Δu term near the boundary or at infinity. Without uniform oscillation control or a comparison lemma that does not rely on Δu = 0, the self-improvement to limsup = 1 (or finite) does not follow from the stated assumptions; this is load-bearing for the claim that the result holds without harmonicity.
[Self-improvement argument (Section 4)] The derivation of the iterated-log estimate appears to proceed by direct iteration on the gradient bounds, but no error term or remainder estimate is supplied to show that the Laplacian contribution remains lower order uniformly. This leaves open whether the precise LIL constant is recovered or only a weaker bound is obtained.

minor comments (2)

[Title] The title misspells 'logarithm' as 'logaritm'.
[Abstract] The abstract is terse and does not indicate the structure of the proof or the precise form of the size conditions, which reduces readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The points raised concern the explicit control of non-harmonic terms in the iteration and the uniformity of remainder estimates. We address each major comment below and will incorporate additional lemmas and estimates in the revised version to strengthen the presentation.

read point-by-point responses

Referee: [Main proof (Section 3)] The argument that the size bounds on |∇u| and |∇Δu| close the LIL iteration for non-harmonic functions lacks explicit absorption estimates for the ∇Δu term near the boundary or at infinity. Without uniform oscillation control or a comparison lemma that does not rely on Δu = 0, the self-improvement to limsup = 1 (or finite) does not follow from the stated assumptions; this is load-bearing for the claim that the result holds without harmonicity.

Authors: We agree that the absorption estimates for the ∇Δu term should be stated more explicitly. The current argument in Section 3 integrates the gradient bounds along vertical rays in the half-space and uses the hypothesis on |∇Δu| to control the second-order contribution via a direct integration-by-parts identity that does not invoke Δu = 0. Nevertheless, a dedicated comparison or oscillation lemma is absent. We will add a new lemma (Lemma 3.4 in the revision) that furnishes uniform oscillation control on balls tangent to the boundary, relying only on the given size conditions on |∇u| and |∇Δu|. This lemma will be proved by a standard covering argument and will close the iteration without harmonicity assumptions. revision: yes
Referee: [Self-improvement argument (Section 4)] The derivation of the iterated-log estimate appears to proceed by direct iteration on the gradient bounds, but no error term or remainder estimate is supplied to show that the Laplacian contribution remains lower order uniformly. This leaves open whether the precise LIL constant is recovered or only a weaker bound is obtained.

Authors: The iteration in Section 4 is constructed so that the term arising from ∇Δu is absorbed into the lower-order error because the hypothesis assumes |∇Δu| = o(r^{-1} (log log r)^{-1/2}) in the relevant scaling. We concede that an explicit remainder estimate is not written out. In the revision we will insert a short calculation (displayed equation (4.7)) that quantifies the accumulated error from the Laplacian term and verifies it is o((log log r)^{-1/2}) uniformly on the sequence of scales used for the limsup. This will confirm that the precise constant 1 is recovered under the stated assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is a direct proof under independent size assumptions

full rationale

The paper states that it proves a version of the Law of the Iterated Logarithm for smooth functions in the upper half-space and derives self-improvement growth properties as a consequence of stated size conditions on the gradient and gradient of the Laplacian. These conditions are presented as inputs that lead to the LIL conclusion and its applications in non-harmonic settings. No equations, definitions, or self-citations in the abstract reduce the claimed result to a fitted parameter, self-referential definition, or load-bearing prior result by the same authors. The central claim remains a mathematical extension with the size bounds serving as external assumptions rather than outputs of the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5572 in / 965 out tokens · 60731 ms · 2026-05-20T03:55:23.478701+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … lim sup |u(x,y) − ∫ t Δu dt| / sqrt(Ψ(y) log log Ψ(y)) ≤ C a.e., proved by transferring to dyadic martingale {T_n} whose increments are bounded by ψ(2^{-k}) and quadratic variation controlled by ∫ ψ²(t)/t dt

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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M. J. Gonz´ alez, P. Koskela,Radial growth of solutions to the Poisson equation, Com- plex Variables Theory Appl. 46 (2001) no.1, 59-72

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J. B. Garnett, D. E. Marshall,Harmonic measure, Cambridge Univ. Press, (2005)

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M. J. Gonz´ alez, A. Nicolau,Multiplicative square functions, Rev. Mat. Iberoameri- cana, 20 (2004) no.3, 673-736

work page 2004
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M. J. Gonz´ alez, J. G. Llorente, P. Koskela, A. Nicolau,Distributional inequalities for non-harmonic functions, Indiana Univ. Math. J. 52 (2003) no.1, 196-226

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Hedenmalm, B

H. Hedenmalm, B. Korenblum, K. Zhu,Theory of Bergman spaces, Springer-Verlag, (2000)

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Ivrii, A

O. Ivrii, A. Nicolau,Analytic mappings of the unit disk with bounded compression, https://doi.org/10.48550/arXiv.2507.15200

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W. F. Stout,A martingale analogue of Kolmogorov’s Law of the Iterated Logartithm. Z. Wahr. verw. Geb. 15(1970), 279-290. J. G. Llorente: Departamento de An ´alisis Matem´atico y Matem´atica Apli- cada. Facultad de Matem´aticas. Universidad Complutense de Madrid. Email address:josgon20@ucm.es A. Nicolau: Departament de Matem`atiques, Universitat Aut`onoma ...

work page 1970

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J. M. Anderson, J. Clunie, Ch. Pommerenke,On Bloch functions and normal func- tions, J. Reine Angew. Math. 270 (1974), 12-37

work page 1974

[2] [2]

Ba˜ nuelos, C

R. Ba˜ nuelos, C. N. Moore,Probabilistic behaviour of harmonic functions, Progress in Mathematics, 175. Birkh¨ auser (1999)

work page 1999

[3] [3]

P. L. Duren, A. Schuster,Bergman spaces, Mathematical Surveys and Monographs, American Mathematical Society, (2004)

work page 2004

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M. J. Gonz´ alez, P. Koskela,Radial growth of solutions to the Poisson equation, Com- plex Variables Theory Appl. 46 (2001) no.1, 59-72

work page 2001

[5] [5]

J. B. Garnett, D. E. Marshall,Harmonic measure, Cambridge Univ. Press, (2005)

work page 2005

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M. J. Gonz´ alez, A. Nicolau,Multiplicative square functions, Rev. Mat. Iberoameri- cana, 20 (2004) no.3, 673-736

work page 2004

[7] [7]

M. J. Gonz´ alez, J. G. Llorente, P. Koskela, A. Nicolau,Distributional inequalities for non-harmonic functions, Indiana Univ. Math. J. 52 (2003) no.1, 196-226

work page 2003

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Hedenmalm, B

H. Hedenmalm, B. Korenblum, K. Zhu,Theory of Bergman spaces, Springer-Verlag, (2000)

work page 2000

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Ivrii, A

O. Ivrii, A. Nicolau,Analytic mappings of the unit disk with bounded compression, https://doi.org/10.48550/arXiv.2507.15200

work page doi:10.48550/arxiv.2507.15200

[10] [10]

J. G. Llorente,Boundary values of harmonic Bloch functions in lipschitz domains: a martingale approach, Potential Anal. 9 (1998) no.3, 229-260

work page 1998

[11] [11]

N. G. Makarov,On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), 369-384

work page 1985

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N. G. Makarov,Probability methods in the theory of conformal mappings, (Russian) Algebra i Analiz 1(1989), no.1, 3-59; translation in Leningrad Math. J. 1(1990), no.1, 1-56

work page 1989

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Pommerenke.Boundary behaviour of conformal maps, Springer-Verlag (1992)

Ch. Pommerenke.Boundary behaviour of conformal maps, Springer-Verlag (1992)

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Shiryaev,Probability, Springer (1991)

A.H. Shiryaev,Probability, Springer (1991)

work page 1991

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W. F. Stout,A martingale analogue of Kolmogorov’s Law of the Iterated Logartithm. Z. Wahr. verw. Geb. 15(1970), 279-290. J. G. Llorente: Departamento de An ´alisis Matem´atico y Matem´atica Apli- cada. Facultad de Matem´aticas. Universidad Complutense de Madrid. Email address:josgon20@ucm.es A. Nicolau: Departament de Matem`atiques, Universitat Aut`onoma ...

work page 1970