On theorems of Chernoff and Ingham on the Heisenberg group

Jayanta Sarkar; Pritam Ganguly; Sayan Bagchi; Sundaram Thangavelu

REVIEW 3 major objections 3 minor 33 references

On theorems of Chernoff and Ingham on the Heisenberg group

T0 review · 3 major / 3 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The paper proves a Chernoff-type quasi-analyticity theorem for the sublaplacian on the Heisenberg group and uses it to prove an Ingham-type Fourier decay theorem.

desk verdict A well-motivated Heisenberg-group extension with a plausible Chernoff theorem, but the main Ingham theorem is not proved: the required sequence ρ_j in §4.2 cannot exist. read the letter →

arxiv 2009.14230 v2 pith:DIQF3GOC submitted 2020-09-29 math.FA math.CA

Sayan Bagchi , Pritam Ganguly , Jayanta Sarkar , Sundaram Thangavelu This is my paper · ORCID

classification math.FAmath.CA MSC 43A8022E2533C4526E1046E35

keywords Heisenberggroupsublaplacianquasi-analyticitySobolevspacesChernoff'stheoremIngham'sStieltjesvectorsFouriertransform

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 claims an analogue of Chernoff's quasi-analyticity theorem for the sublaplacian on the Heisenberg group and uses it to prove a version of Ingham's theorem for the group Fourier transform. In the Chernoff half, angular-symmetric smooth functions whose iterates $L^m f$ grow slowly enough, so that $\sum_m \|L^m f\|_2^{-1/2m}$ diverges, cannot vanish with all derivatives at one point unless they vanish everywhere. In the Ingham half, decay of $\hat f(\lambda)^*\hat f(\lambda)$ against the operator exponential $e^{-2\Theta(\sqrt{H(\lambda)})\sqrt{H(\lambda)}}$ is claimed to be possible, for a compactly supported nonzero $f$, exactly when $\int_1^\infty \Theta(t)/t\,dt$ converges, under the same angular-symmetry hypothesis on $f$. This matters because it transfers sharp decay thresholds from Euclidean Fourier analysis to a noncommutative, operator-valued setting, and it makes the sublaplacian itself the generator of the quasi-analytic class.

What carries the argument

The load-bearing mechanism is the Stieltjes-vector criterion for essential self-adjointness, together with the Fourier correspondence $\widehat{Lf}(\lambda)=\hat f(\lambda)H(\lambda)$ on the Heisenberg group. A vector $f$ with $\sum_m \|L^m f\|_2^{-1/2m}=\infty$ is a Stieltjes vector for $L$; a theorem of Chernoff says that if the span of such vectors is dense, the symmetric operator is essentially self-adjoint. The paper proves that the sublaplacian restricted to smooth functions vanishing at the origin is not essentially self-adjoint, so any $f$ of the stated kind would make the span dense, a contradiction. For the Ingham half, the same Fourier correspondence identifies decay of $\hat f(\lambda)$ with decay of the spectral coefficients against the Hermite operator $H(\lambda)$, and sharp Laguerre-function estimates (Lemmas 2.1 and 4.4) control those coefficients for the specially built compactly supported functions obtained as infinite convolutions of ball characteristic functions.

What would settle it

Check whether any decreasing sequence $\rho_j$ can satisfy $\rho_j \ge c_n^2 e^{2\Theta(j)}/j$ for all $j$ and still have $\sum_j \rho_j<\infty$. Because $\Theta$ is nonnegative, the lower bound forces $\rho_j \ge c_n^2/j$, so the sum diverges for every admissible $\Theta$; this directly contradicts the compact-support requirement in Theorem 4.3 and shows the sufficiency proof in Section 4.2 cannot be completed as written.

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Extended reading notes

Core claim

The paper's central claim is a pair of theorems. Theorem 1.5 is an analogue of Chernoff's theorem for the sublaplacian $L$ on $H^n$: for a smooth angular-symmetric function $f$, meaning $f(\rho\omega\sqrt{\sin\theta},\rho^2\cos\theta)=f(\rho\omega,0)$ in Heisenberg coordinates, if $L^m f\in L^2(H^n)$ for all $m$ and $\sum_{m=1}^\infty \|L^m f\|_2^{-1/2m}=\infty$, then $f$ and all its partial derivatives vanishing at the origin imply $f\equiv 0$. Theorem 1.3 is an Ingham-type theorem: for an even, decreasing, nonnegative $\Theta$ with $\Theta(\lambda)\to 0$, convergence of $\int_1^\infty \Theta(t)\,t^{-1}dt$ is claimed to be exactly the condition under which a nonzero compactly supported continuous $f$ can have $\hat f(\lambda)^*\hat f(\lambda)\le C e^{-2\Theta(\sqrt{H(\lambda)})\sqrt{H(\lambda)}}$, with the necessity direction requiring the same angular-symmetry hypothesis and vanishing near $0$. The paper notes explicitly that removing the angular-symmetry hypothesis would need a more general Chernoff theorem that is not yet available.

Load-bearing premise

The sufficiency direction of Theorem 1.3 rests on the assumption that one can choose a decreasing sequence of positive radii $\rho_j$ with $\rho_j \ge c_n^2 e^{2\Theta(j)}/j$ for every $j$ and also $\sum_j \rho_j<\infty$; the nonnegativity of $\Theta$ makes this impossible, since each term is at least $c_n^2/j$.

Editorial extensions

If this is right

Corollary 1.6 gives an $L^2$ Denjoy--Carleman theorem on $H^n$: for a log-convex sequence $\{M_k\}$ with $\sum_k M_k^{-1/2k}=\infty$, every angular-symmetric smooth function satisfying $\|L^k f\|_2 \le M_k\lambda^k$ is quasi-analytic.
For an angular-symmetric $f$ with $\hat f(\lambda)^*\hat f(\lambda)\le C e^{-2\Theta(\sqrt{H(\lambda)})\sqrt{H(\lambda)}}$ and with $\int_1^\infty \Theta(t)/t\,dt=\infty$, vanishing on a neighbourhood of $0$ forces $f\equiv 0$.
If Theorem 1.3 is valid, then whenever $\int_1^\infty \Theta(t)/t\,dt<\infty$ there exists a nonzero compactly supported continuous $f$ with the stated operator-valued Fourier decay, giving explicit substitute bump functions in the noncommutative setting.
The angular-symmetry class includes all lifts $f(z,t)=g(\rho\omega)$ of functions on $\mathbb{C}^n$, so the theorem covers a large family beyond radial functions.

Reading between the lines

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

The displayed lower bound $\rho_j \ge c_n^2 e^{2\Theta(j)}/j$ cannot coexist with $\sum_j \rho_j<\infty$ for nonnegative $\Theta$, so a repaired sufficiency proof would need a genuinely different construction, not merely a better choice of constants.
Because the angular-symmetry hypothesis is used only to separate points by the family $\{R_\sigma \delta_r f\}$, groups with richer symmetry actions may admit the analogous Chernoff theorem without any extra assumption; the present obstruction is a missing separation argument, not a failure of the Stieltjes-vector method.
The paper's remark that the result should hold without extra symmetry suggests the correct general theorem would require a genuinely $H^n$-valued separation step; finding such a step is a concrete open problem.
The sharp Laguerre bounds in Lemma 2.1 could be pushed to quantitative support-radius estimates, so the existence theorem might yield explicit size bounds for the supporting bumps in terms of $\Theta$.

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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

Here is the honest take. The paper has a real new idea: measure Fourier decay on the Heisenberg group through the Hermite semigroup rather than just Hilbert-Schmidt norms, and it proves a Chernoff-type theorem for the sublaplacian, albeit for functions independent of the angular variable θ in Heisenberg coordinates. The Chernoff proof in Section 3 is a clean application of Stieltjes-vector theory and may well be correct.

The problem is the Ingham theorem. The sufficiency direction in Section 4.2 requires a decreasing sequence ρ_j with ρ_j ≥ c_n^2 e^{2Θ(j)}/j for all j and ∑ρ_j < ∞. Since Θ ≥ 0, e^{2Θ(j)} ≥ 1, so the lower bound is at least c_n^2/j. The harmonic series diverges. No such sequence exists. This is not a technicality: the summability of ρ_j is what makes the limit function F compactly supported in Theorem 4.3, while the lower bound is what gives the decay estimate in Theorem 4.5. The two requirements are contradictory. So the sufficiency direction of Theorem 1.3 is not proved.

The necessity proof in Section 4.3 uses this same construction with Ψ(t) = (1+|t|)^{-1/2} and approximate identities, so it inherits the gap. The convolution argument for h = f*F cannot get off the ground without a valid F. There is also a smaller related issue: the estimate ∑ ||L^m h||^{-1/2m} = ∞ depends on the decay of F, which is exactly what the broken construction should have provided.

If the authors can replace the impossible sequence with a different construction, the Ingham theorem might be salvageable. As written, the central claim is unproven. The Chernoff part is worth something on its own.

Who is this for? Researchers working on uncertainty principles on nilpotent groups. The topic matters and the Chernoff part appears sound, but the Ingham part needs serious repair. I would not accept the paper as is, but it deserves a serious referee.

Referee Report

3 major / 3 minor

Summary. The paper proves two theorems on the Heisenberg group H^n. Theorem 1.5 is a Chernoff-type quasi-analyticity result for the sublaplacian L: under an angular-independence condition in Heisenberg coordinates, if L^m f is in L^2 for all m and the Stieltjes-type series diverges, then vanishing of f and all its derivatives at the origin forces f=0. The proof uses essential self-adjointness criteria and Stieltjes vectors. Theorem 1.3 is an Ingham-type uncertainty theorem for the group Fourier transform: under an integral condition on an even decreasing function Θ, the paper constructs a nonzero compactly supported continuous f whose Fourier transform satisfies the operator decay estimate (1.3), and conversely, under an additional angular-independence hypothesis, such decay together with vanishing near the origin forces the integral condition. The Ingham theorem is proved by constructing F as an infinite convolution of box-type functions and estimating the resulting Laguerre coefficients, then applying the Chernoff theorem after a smoothing step.

Significance. If the main results were correct, they would be a valuable extension of the Ingham-type uncertainty principles proved by Bhowmik-Ray-Sen and Bhowmik-Pusti-Ray to the operator-valued Fourier transform on the Heisenberg group, and the Chernoff theorem for the sublaplacian would be a useful tool in its own right. The paper is clearly organized, and the proof of Theorem 1.5 is substantive: it uses the Stieltjes-vector framework and contains detailed estimates with the matrix coefficients e^λ_{α,β}. The Laguerre-coefficient estimates in Lemma 4.4 are also presented in a transparent way. However, the central sufficiency construction in Section 4.2 rests on an impossible choice of sequences, and this invalidates the proof of Theorem 1.3. Because the main theorem is not established, the paper cannot be accepted in its current form.

major comments (3)

[§4.2, sequence choice before Lemma 4.4] The assertion that one can choose a decreasing summable sequence ρ_j with ρ_j ≥ c_n^2 e^{2Θ(j)}/j is false. Since Θ is nonnegative, e^{2Θ(j)} ≥ 1 for every j, so the lower bound forces ρ_j ≥ c_n^2/j for all j. Hence ∑ρ_j ≥ c_n^2 ∑j^{-1} = ∞, contradicting the required convergence of ∑ρ_j. This is not a cosmetic issue: the summability of ρ_j is exactly what makes the support of the limiting function F contained in B(0, a∑ρ_j + c∑τ_j) in Theorem 4.3, while the lower bound is used in Theorem 4.5 to force the factor e^{-2Θ(√(2k+n)|λ|)√(2k+n)|λ|}. The two requirements on ρ_j are contradictory, so the sufficiency direction of Theorem 1.3 is not proved.
[Theorem 4.6] The necessity proof inherits the same obstruction. To treat the general case, the proof invokes the construction with Ψ(y)=(1+|y|)^{-1/2} and asserts the existence of a compactly supported F satisfying the decay estimate with Ψ. Since Ψ is nonnegative, the required lower bound would again be ρ_j ≥ c_n^2 e^{2Ψ(j)}/j ≥ c_n^2/j, so no decreasing summable sequence ρ_j exists. Consequently the smoothing step involving h = f * F and the functions F_ε is unsupported.
[Theorem 4.6, general case] Even if the sequence problem were repaired, the step "By the previous part of the theorem it follows that ∑‖L^m(f*F_ε)‖^{-1/(2m)} = ∞" applies the special case that was proved under the explicit hypothesis Θ(λ) ≥ 2|λ|^{-1/2}. For Φ = Θ + εΨ(εy), the available lower bound is only of the form Φ(y) ≥ c_ε |y|^{-1/2} with c_ε depending on ε, and for small ε this does not satisfy the stated hypothesis. The argument would need a separate proof of the Stieltjes-vector property for f*F_ε under a lower bound with constant ε; as written, this is a gap in the reduction to the special case.

minor comments (3)

[Theorem 4.3 and Theorem 1.3] Theorem 4.3 proves convergence of the convolutions G_k to F in L^1 and L^2, but Theorem 1.3 asserts that the constructed function is continuous. Continuity would follow from the uniform estimate (4.7) if the sequences are chosen so that ∑τ_j^2 < ∞ and ∑ρ_j < ∞, but this detail is not stated in the construction.
[Proof of Theorem 4.6] The sentence "By Theorem 4.3 we can construct a radial function F ∈ L^2(H^n) supported in B(0,δ/2) such that ..." cites the wrong theorem: Theorem 4.3 only constructs the compactly supported L^2 limit, while the decay estimate for its Fourier transform is the content of Theorem 4.5.
[§4.2, definitions of f_j and g_j] The text says "we define functions f_j on C^n and τ_j on R" but the second function is meant to be g_j(t); this is a typo that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's central claims are forward derivations from background facts; the known gap is a mathematical error, not a self-referential reduction.

full rationale

The derivation chain is forward and self-contained: Theorem 1.5 (Chernoff) is proved in Section 3 using standard material on Stieltjes vectors and essential self-adjointness, and Theorem 1.3 (Ingham) is then derived from it in Section 4. The sufficiency construction (Theorems 4.3 and 4.5) builds compactly supported functions by convolution from scratch; the necessity part (Theorem 4.6) uses the Chernoff theorem proved earlier plus the constructed F. Citations, including [13] and [31] by the authors, supply standard Fourier-transform identities and Laguerre coefficient formulas rather than the target conclusions. The theorem is explicitly stated with an extra angular assumption, and the paper openly notes that removing it requires a yet-unproved general Chernoff theorem; a conditional limitation is not a circular step. The reviewer-flagged impossibility of choosing a summable decreasing sequence rho_j with rho_j >= c_n^2 e^{2Theta(j)}/j is a genuine correctness gap in the proof of the sufficiency direction, but it is a mathematical error, not a definitional or self-citational circularity. No step in the paper reduces a claimed prediction to an input by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The proof uses standard harmonic analysis background plus the paper's own Chernoff theorem. The only hand-chosen quantities are the sequences ρ_j and τ_j and normalization constants; the ρ_j choice is the source of the critical error. No new entities are introduced.

free parameters (3)

decreasing sequences ρ_j and τ_j = chosen with ∑ρ_j<∞, ∑τ_j<∞ and ρ_j≥c_n^2e^{2Θ(j)}/j
The support of the constructed function F and the Fourier decay estimate both depend on these sequences; the lower bound on ρ_j is impossible to reconcile with summability.
constant c_n = chosen later, depends only on n
Appears in Lemma 4.4 and in the sequence lower bound; used to absorb constants in estimates.
constant a = chosen so ||f_j||_1=1
Normalization of the building blocks in the convolution construction; harmless but hand-chosen.

assumptions (5)

standard math Schrödinger representations and Plancherel theorem for H^n
Background for operator-valued Fourier transform, Section 2.1.
standard math Sharp Laguerre function estimates (Lemma 2.1)
Quoted from [29, Lemma 1.5.3] and used to estimate Fourier coefficients in Lemma 4.4.
standard math Stieltjes vector theorem (Theorem 3.1) and essential self-adjointness characterization
Core tool for the Chernoff theorem proof, Section 3.
standard math Hausdorff measure comparison (Theorem 4.2)
Used in the L2 convergence proof of the convolution sequence.
domain assumption Extra symmetry assumption f(ρω√sinθ, ρ²cosθ)=f(ρω,0) for functions in the Chernoff theorem and the necessity part of Ingham theorem
Limits the class of functions; the authors state they believe it is unnecessary.

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Pith. "Pith review of On theorems of Chernoff and Ingham on the Heisenberg group." pith.science (2026). https://pith.science/paper/DIQF3GOC

@misc{pith2026200914230,
  author       = {Pith},
  title        = {Pith review of: On theorems of Chernoff and Ingham on the Heisenberg group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DIQF3GOC}},
  note         = {Machine review of arXiv:2009.14230}
}

read the original abstract

We prove an analogue of Chernoff's theorem for the sublaplacian on the Heisenberg group and use it prove a version of Ingham's theorem for the Fourier transform on the same group.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Ahlfors, Complex Analysis: An introduction to the theory of analytic functions of one complex variable, 3rd Edition, McGraw-Hill, Inc

Lars V. Ahlfors, Complex Analysis: An introduction to the theory of analytic functions of one complex variable, 3rd Edition, McGraw-Hill, Inc

work page

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Bhowmik, S

M. Bhowmik, S. K. Ray, and S.Sen, Around theorems of Ingham-t ype regarding decay of Fourier trans- form on Rn,Tn and two step nilpotent Lie Groups, Bull. Sci. Math , 155 (2019) 33-73

work page 2019

[3] [3]

Bhowmik, S

M. Bhowmik, S. Pusti, and S. K. Ray, Theorems of Ingham and Che rnoﬀ on Riemannian symmetric spaces of noncompact type, Journal of Functional Analysis , Volume 279, Issue 11 (2020)

work page 2020

[4] [4]

Bhowmik and S

M. Bhowmik and S. Sen, Uncertainty principles of Ingham and Paley -wiener on semisimple Lie groups, Israel Journal of Mathematics 225 (2018),193-221

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[5] [5]

Bochner and A

S. Bochner and A. E. Taylor, Some theorems on quasi-analyticity for functions of several variables, Amer. J. Math. ,61 (1939), no-2,303-329. CHERNOFF AND INGHAM ON THE HEISENBERG GROUP 25

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Bochner, Quasi-analytic functions, Laplace operator, posit ive kernels Ann

S. Bochner, Quasi-analytic functions, Laplace operator, posit ive kernels Ann. of Math. (2) , 51 (1950), 68-91. MR0032708 (11,334g)

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R.Chernoﬀ, Quasi-analytic vectors and quasi-analytic functio ns

P. R.Chernoﬀ, Quasi-analytic vectors and quasi-analytic functio ns. Bull. Amer. Math. Soc. 81 (1975), 637-646

work page 1975