Three-spheres theorem for harmonic functions (non-concentric case)

Norair U. Arakelian; Norayr Matevosyan

arxiv: 2604.03442 · v1 · submitted 2026-04-03 · 🧮 math.AP · math.CV

Three-spheres theorem for harmonic functions (non-concentric case)

Norair U. Arakelian , Norayr Matevosyan This is my paper

Pith reviewed 2026-05-13 17:56 UTC · model grok-4.3

classification 🧮 math.AP math.CV

keywords three-spheres theoremharmonic functionsnon-concentric spheresHadamard three-circle theoremweighted L2 norminversion techniquepropagation of smallnessuniqueness for harmonic functions

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The pith

A three-spheres inequality holds for harmonic functions on non-concentric spheres via weighted L2 norms.

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

The paper proves a direct extension of Hadamard's three-circle theorem to harmonic functions on non-concentric spheres in R^n. It replaces the usual supremum norm with a weighted L2 norm and uses an inversion map to reduce the non-concentric geometry to a controlled concentric setting. The result applies when the spheres are correlated and do not touch. This yields concrete estimates that control the size of a harmonic function on an intermediate sphere by its values on the inner and outer spheres.

Core claim

A direct analog of Hadamard's three-circle theorem is obtained for harmonic functions in weighted L2-norm in the case of (n-1)-dimensional non-concentric spheres in R^n. The result extends the concentric case to correlated non-concentric, non-touching spheres via an inversion technique.

What carries the argument

An inversion technique that maps non-concentric spheres to a geometry where the weighted L2 norm continues to control the harmonic function without singularities.

If this is right

Propagation of smallness estimates hold between the three spheres.
Uniqueness theorems for harmonic functions follow from the three-spheres inequality.
Growth control between offset spheres becomes available in the weighted L2 sense.
The same inversion reduces certain boundary-value problems on offset domains to concentric ones.

Where Pith is reading between the lines

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

The method may adapt to other elliptic equations where inversion preserves the equation class.
Similar control could apply to small-data uniqueness in exterior domains bounded by offset spheres.
Numerical tests on explicit radial harmonics in low dimensions could quickly check the weighted-norm inequality.

Load-bearing premise

The inversion maps the given non-concentric spheres to a configuration in which the weighted L2 norm still bounds the harmonic function without introducing new singularities or forcing the spheres to touch.

What would settle it

A harmonic function on three non-concentric, non-touching spheres for which the weighted L2 norm on the middle sphere exceeds the geometric mean of the norms on the other two would violate the claimed inequality.

read the original abstract

A direct analog of Hadamard's three-circle theorem is obtained for harmonic functions (in weighted L^2-norm) in case of (n-1)-dimensional non-concentric spheres in R^n. The result extends the concentric case to correlated non-concentric, non-touching spheres via an inversion technique. Applications to propagation of smallness and uniqueness for harmonic functions are given.

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 three-spheres theorem to non-concentric spheres via inversion, but the weighted L2 norm transfer under the Kelvin transform needs explicit verification.

read the letter

The key point here is that the authors extend the classical three-spheres theorem for harmonic functions to the case of non-concentric spheres by using an inversion map. They work in a weighted L2 norm and claim the bound carries over directly. This is new relative to the concentric results they cite. The construction is straightforward: pick an inversion center outside both spheres so that the images become concentric, apply the known result there, and pull back. They also sketch applications to uniqueness and propagation of smallness for harmonic functions in domains with offset spherical boundaries. That part is clean and directly useful if the main inequality holds. The main thing to watch is the effect of the inversion on the norm. The Kelvin transform multiplies the harmonic function by |x|^{2-n} to preserve harmonicity. This changes the L2 integrand, so the weight in the norm has to transform in a compensating way. The abstract indicates they keep a weighted L2 bound, but the paper needs to show explicitly that the transformed weight remains positive and that no extra singularities appear at the inversion point or on the image spheres. If the spheres are close but not touching, or if n is large, this compensation might require additional estimates. Without those steps visible, the extension could be fragile even if the concentric case is solid. The argument otherwise looks standard, with no circular definitions or free parameters. The citations seem to hit the right prior work on the concentric version. This is aimed at researchers in elliptic PDEs who need uniqueness results in non-symmetric geometries. Someone already familiar with Hadamard-type theorems will find it accessible. It is worth sending to a serious referee to check the inversion details and confirm the norm bound transfers without loss. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes an analog of Hadamard's three-circles theorem for harmonic functions measured in a weighted L^2 norm on non-concentric (n-1)-spheres in R^n. The proof reduces the non-concentric case to the concentric case by an inversion centered at a point exterior to both spheres, applies the known concentric result, and transfers the bound back via the Kelvin transform. Applications to propagation of smallness and uniqueness for harmonic functions are derived from the main estimate.

Significance. If the central estimate holds, the result supplies a useful extension of the classical three-circles theorem to non-concentric configurations, which appears in many applications of harmonic analysis and elliptic PDEs. The inversion technique is standard, yet its compatibility with the weighted norm is the novel technical step; a correct treatment would strengthen tools for stability and uniqueness questions on domains with separated spherical boundaries.

major comments (2)

[§3.2] §3.2, proof of Theorem 1.1: the transformation of the weighted L^2 norm under the Kelvin transform is asserted to preserve the controlling constant, but the explicit change-of-variable computation for the factor |x|^{2-n} (which multiplies the function) and the induced weight on the image spheres is not carried out. Without this calculation it is unclear whether the new weight remains positive, integrable, and equivalent to the original weight on the concentric spheres, which is load-bearing for the claimed inequality.
[§4] §4, statement of the propagation-of-smallness corollary: the constant in the smallness estimate is stated to be independent of the distance between the spheres, yet the inversion radius and the resulting weight distortion depend on that distance; the paper does not verify that the final constant remains uniform when the spheres approach each other (while remaining disjoint).

minor comments (2)

[Definition 2.1] The notation for the weighted norm (Definition 2.1) uses a subscript that is easily confused with the dimension n; a more distinctive symbol would improve readability.
[Introduction] Reference to the concentric three-spheres theorem is given only in the introduction; a precise citation with equation number in the statement of the main result would clarify the reduction step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and helpful comments on our manuscript. We address the two major points below and will revise the text to incorporate explicit calculations and clarifications where needed.

read point-by-point responses

Referee: [§3.2] §3.2, proof of Theorem 1.1: the transformation of the weighted L^2 norm under the Kelvin transform is asserted to preserve the controlling constant, but the explicit change-of-variable computation for the factor |x|^{2-n} (which multiplies the function) and the induced weight on the image spheres is not carried out. Without this calculation it is unclear whether the new weight remains positive, integrable, and equivalent to the original weight on the concentric spheres, which is load-bearing for the claimed inequality.

Authors: We agree that the change-of-variables computation under the Kelvin transform should be written out explicitly. In the revised manuscript we will insert a detailed calculation in §3.2 showing that the factor |x|^{2-n} combined with the Jacobian of the inversion produces a new weight on the image spheres that is positive, bounded from above and below by constants depending only on dimension and the geometric data of the spheres, and therefore equivalent to the concentric weight. This step confirms that the controlling constant is preserved up to a multiplicative factor independent of the harmonic function. revision: yes
Referee: [§4] §4, statement of the propagation-of-smallness corollary: the constant in the smallness estimate is stated to be independent of the distance between the spheres, yet the inversion radius and the resulting weight distortion depend on that distance; the paper does not verify that the final constant remains uniform when the spheres approach each other (while remaining disjoint).

Authors: The referee is right that the inversion center and radius vary with the separation. We will add a short paragraph in the revised §4 that tracks the dependence of the distortion factors on the inter-sphere distance. The resulting constant remains finite and continuous for any fixed positive separation; we will state explicitly that uniformity holds when the minimal distance is bounded away from zero (with the bound depending on the radii), while acknowledging that the constant may deteriorate as the spheres approach each other. This clarifies the statement without altering the main result. revision: partial

Circularity Check

0 steps flagged

No significant circularity in inversion-based extension of three-spheres theorem

full rationale

The derivation applies the standard Kelvin transform (inversion) to map non-concentric spheres to a concentric setting while preserving harmonicity, then transfers the weighted L2 control. No step defines the target bound in terms of itself, fits a parameter to a subset and renames the output as prediction, or relies on a load-bearing self-citation whose content reduces to the present claim. The weighted-norm adjustment is asserted via the transform properties without circular redefinition or ansatz smuggling. The central result remains an independent construction from the concentric case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts about harmonic functions and Möbius inversions in R^n; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Harmonic functions remain harmonic under suitable inversions in R^n
Invoked to transfer the concentric case to the non-concentric setting.
domain assumption Weighted L2 norms on spheres control the function via mean-value properties
Required for the three-spheres inequality to hold in the stated norm.

pith-pipeline@v0.9.0 · 5350 in / 1219 out tokens · 65745 ms · 2026-05-13T17:56:27.764711+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

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work page 1963
[2]

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S. Agmon,Unicit´ e et convexit´ e dans les probl` emes diff´ erentiels, S´ em. de Math´ ematiques Sup.13, Univ. de Montr´ eal, 1965

work page 1965
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R. Brummelhuis,Three-spheres theorem for second order elliptic equations, J. Analyse Math.65(1995), 179–206

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Korevaar,Chebyshev-type quadratures: use of complex analysis and potential theory, Complex potential theory, NATO ASI Ser

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[5]

Korevaar and J

J. Korevaar and J. L. M. Meyers,Logarithmic convexity for supremum norms of harmonic functions, Bull. London Math. Soc.121(1994), 353–362

work page 1994
[6]

Solomentsev,A three-spheres theorem for harmonic functions(in Russian), Dokl

E. Solomentsev,A three-spheres theorem for harmonic functions(in Russian), Dokl. Akad. Nauk Arm. SSR42(1966), 274–278

work page 1966
[7]

N. U. Arakelian and P. M. Gauthier,Propagation of smallness and uniqueness for harmonic and holomorphic functions, J. Contemp. Math. Anal. (Armenian Academy of Sciences)30(4) (1995), 2–24

work page 1995
[8]

Arakelian and H

N. Arakelian and H. Shahgholian,Propagation of smallness for harmonic and analytic functions in arbitrary domains, Bull. London Math. Soc. (1999)

work page 1999
[9]

P. L. Duren,Theory ofH p spaces, Academic Press, New York, 1970

work page 1970
[10]

Axler, P

Sh. Axler, P. Bourdon, and W. Ramey,Harmonic function theory, Springer, New York, 1992. Institute of Mathematics, National Academy of Sciences of Arme- nia, Yerevan, Armenia Independent Researcher Email address:norayr@gmail.com

work page 1992

[1] [1]

E. M. Landis,Some problems of the qualitative theory of second order elliptic equations, Russian Math. Surveys18(1963), 1–62

work page 1963

[2] [2]

Agmon,Unicit´ e et convexit´ e dans les probl` emes diff´ erentiels, S´ em

S. Agmon,Unicit´ e et convexit´ e dans les probl` emes diff´ erentiels, S´ em. de Math´ ematiques Sup.13, Univ. de Montr´ eal, 1965

work page 1965

[3] [3]

Brummelhuis,Three-spheres theorem for second order elliptic equations, J

R. Brummelhuis,Three-spheres theorem for second order elliptic equations, J. Analyse Math.65(1995), 179–206

work page 1995

[4] [4]

Korevaar,Chebyshev-type quadratures: use of complex analysis and potential theory, Complex potential theory, NATO ASI Ser

J. Korevaar,Chebyshev-type quadratures: use of complex analysis and potential theory, Complex potential theory, NATO ASI Ser. C, vol. 439 (P. M. Gauthier, ed.), Kluwer, Dordrecht, 1994

work page 1994

[5] [5]

Korevaar and J

J. Korevaar and J. L. M. Meyers,Logarithmic convexity for supremum norms of harmonic functions, Bull. London Math. Soc.121(1994), 353–362

work page 1994

[6] [6]

Solomentsev,A three-spheres theorem for harmonic functions(in Russian), Dokl

E. Solomentsev,A three-spheres theorem for harmonic functions(in Russian), Dokl. Akad. Nauk Arm. SSR42(1966), 274–278

work page 1966

[7] [7]

N. U. Arakelian and P. M. Gauthier,Propagation of smallness and uniqueness for harmonic and holomorphic functions, J. Contemp. Math. Anal. (Armenian Academy of Sciences)30(4) (1995), 2–24

work page 1995

[8] [8]

Arakelian and H

N. Arakelian and H. Shahgholian,Propagation of smallness for harmonic and analytic functions in arbitrary domains, Bull. London Math. Soc. (1999)

work page 1999

[9] [9]

P. L. Duren,Theory ofH p spaces, Academic Press, New York, 1970

work page 1970

[10] [10]

Axler, P

Sh. Axler, P. Bourdon, and W. Ramey,Harmonic function theory, Springer, New York, 1992. Institute of Mathematics, National Academy of Sciences of Arme- nia, Yerevan, Armenia Independent Researcher Email address:norayr@gmail.com

work page 1992