Sharp estimates for Jacobi heat kernels in double conic domains

Dawid Hanrahan

REVIEW 2 major objections 4 minor 15 references

Sharp estimates for Jacobi heat kernels in double conic domains

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves genuinely sharp two-sided estimates for even and odd Jacobi heat kernels on double cones, their surfaces, and hyperbolic counterparts, with explicit formulas.

desk verdict The central 'After noticing' comparability in Section 3 is non-uniform, so the main sharp estimates for the double cone and hyperboloid are not established, though the paper is otherwise well structured and likely fixable. read the letter →

arxiv 2411.15793 v1 pith:NNSCMOXL submitted 2024-11-24 math.AP

Dawid Hanrahan This is my paper · ORCID

classification math.AP MSC 35K0833C50

keywords Jacobiheatkerneldoubleconehyperboloidparaboloidsharpestimatesorthogonalpolynomialsbounds

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

This paper seeks genuinely sharp two-sided estimates for the even and odd Jacobi heat kernels on the surface and interior of the multidimensional double cone, and for the even kernel on the hyperboloid and its surface. Genuinely sharp means the kernel is trapped between constant multiples of one explicit expression, uniformly in the two points and in the heat time $\tau$. The main results, Theorems 1.6 and 1.12, give such expressions: a Gaussian in $\arccos(I_1+I_2)$ (with an extra variable $I_3$ in the solid case), multiplied by a power of $\tau$ and by factors $(I_2\vee\tau)^{-\gamma}$ and $(I_3\vee\tau)^{-\mu}$. The odd kernels are obtained by multiplying by $st$, and the hyperbolic results follow by a change of variables. The paraboloid cases are left open, because their orthogonal polynomials are not eigenspaces of a single diffusion operator with degree-only eigenvalues.

What carries the argument

The carrying device is an integral representation: the double-cone heat kernel is written as an integral, against a $\beta$-type measure $d\Pi_{\gamma-\frac12}(v)$ (and a second such integral in the solid case), of the classical one-dimensional Jacobi heat kernel evaluated at a linear argument $\xi(v)=I_1+vI_2$ or $\xi(u,v)=I_1+vI_2+uI_3$. Sharp interval estimates for that kernel, a monotonicity reduction of the integration range, and a boundary evaluation lemma for integrals of $\exp(-\arccos^2(A+Bw)/(4\tau))$ convert the integral into the closed-form boundary expression stated in the theorems.

What would settle it

Set $d=2$, $\gamma=1$, take antipodal unit vectors $x=-y$ on $V_0^3$ with $t=s=\sqrt{1-\varepsilon}$ and $\tau=\varepsilon^{3/4}$; then $I_1=-1$, $I_2=\varepsilon$, and $\pi-\arccos(I_1+I_2)\asymp\sqrt{\varepsilon}$, so the asserted comparability would claim $(\sqrt{\varepsilon})^{-1}\asymp(\varepsilon^{3/4})^{-1}$, which fails as $\varepsilon\to0$. Evaluating the two sides numerically at these parameters settles the uniformity question.

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Extended reading notes

Core claim

On the surface of the double cone $V_0^{d+1}$, the even Jacobi heat kernel $h^\mathrm{E}_\tau$ is comparable, uniformly for $\tau\in(0,1]$, to $\tau^{-d/2}(\pi-\arccos(I_1+I_2)+\tau)^{-\gamma-d/2+1/2}(I_2\vee\tau)^{-\gamma}\exp(-\arccos^2(I_1+I_2)/(4\tau))$, and for $\tau>1$ it is comparable to $1$. On the solid double cone, the even kernel is comparable to $\tau^{-d/2-1/2}(\tau+\pi-\arccos(I_1+I_2+I_3))^{-\gamma-\mu-d/2}(I_2\vee\tau)^{-\gamma}(I_3\vee\tau)^{-\mu}\exp(-\arccos^2(I_1+I_2+I_3)/(4\tau))$. The odd kernels obey the same bounds with an additional factor $st$, and the hyperbolic analogues follow by replacing $t,s$ with $\sqrt{t^2-\rho^2},\sqrt{s^2-\rho^2}$.

Load-bearing premise

The load-bearing step is the assertion, made without proof in Section 3, that $\big(I_2/(\pi-\arccos(I_1+I_2))+\tau\big)^{-\gamma}$ is uniformly comparable to $(I_2\vee\tau)^{-\gamma}$; near the boundary $t,s\to1$ with $I_1+I_2\to-1$ the two sides differ by an unbounded factor for intermediate $\tau$.

Editorial extensions

If this is right

For $\tau\in(0,1]$, the even kernel on $V_0^{d+1}$ decays like $\tau^{-d/2}(I_2\vee\tau)^{-\gamma}$ times the Gaussian $\exp(-\arccos^2(I_1+I_2)/(4\tau))$, uniformly in both points.
The odd kernels on both double cones are bounded by the same expressions with an extra factor $|st|$, so they vanish near the cone tips.
The hyperboloid estimates are the double-cone estimates after the change $t\mapsto\sqrt{t^2-\rho^2}$, $s\mapsto\sqrt{s^2-\rho^2}$.
For $\tau>1$, every treated kernel is uniformly comparable to $1$.
The paraboloid settings are not covered: the diffusion operator there has eigenvalues depending on both the total degree $n$ and the internal degree $m$.

Reading between the lines

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

Beyond the paper: the closed forms suggest a geometric reading of $\arccos(I_1+I_2)$ (and $\arccos(I_1+I_2+I_3)$) as a distance-like angle between the two points, so on these domains the short-time heat kernel has the familiar Gaussian shape with power-law corrections whose exponents are the weight parameters $\gamma$ and $\mu$.
Beyond the paper: the same integral-reduction scheme should apply to any domain whose reproducing kernel is a one- or two-fold integral of a univariate special function with a linear argument and whose eigenvalues depend only on the total degree; the paraboloid fails the second condition.
Beyond the paper: one testable extension is to check numerically whether the stated bound remains uniform when one point approaches the cone boundary while the other is held fixed and $\tau$ scales with the boundary distance; the current proof leaves that regime open.

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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

Let me be direct: the central theorems are not proven as stated. The derivation is mostly clean, and the author honestly flags the paraboloid limitations, but the 'After noticing' comparability in Section 3 is asserted without proof and, as written, false uniformly. Near the outer boundary, take t=s=1−ε, x=t e1, y=−s e1, τ=ε^{3/4}. Then I2≈2ε, π−arccos(I1+I2)≈2√(2ε), so I2/(π−ψ)≈√(ε/2). For γ>0, (I2/(π−ψ)+τ)^−γ≈ε^{−γ/2} while (I2∨τ)^−γ=ε^{−3γ/4}; the ratio tends to 0. So the claimed equivalence in Theorem 1.6 fails. The same false step appears in the proof of Theorem 1.12 when asserting π−arccosξ(1,v) is comparable to a constant; that fails in the same regime. Every corollary inherits the gap.

What is good: the paper is clearly written, the reduction to the Jacobi interval kernel via Lemmas 2.1 and 2.2 is correct, the parameter bookkeeping is careful, and the negative observations about the paraboloid are honest and useful. The architecture is exactly what NSS21 does, applied to Xu's domains, and the author credits that properly.

The flaw is load-bearing, not a minor typo. But it may be repairable: perhaps the boundary factor should keep the I2/(π−arccos) term, or the stated comparability is only valid in a different range. A referee should ask for a corrected statement or a rigorous proof of that step. As it stands, the sharp estimates for the double cone and hyperboloid are not established.

I would recommend sending this to peer review rather than desk-rejecting: the error is subtle, not sloppiness, and a motivated referee or author could fix it. But I would not cite the main theorems until that step is repaired.

Referee Report

2 major / 4 minor

Summary. The paper studies even and odd Jacobi heat kernels associated with Xu's orthogonal polynomial frameworks on the surface and solid double cone, the hyperboloid, and the paraboloid. The main results (Theorems 1.6 and 1.12, and Corollaries 1.7, 1.13, 1.17, 1.21) assert explicit two-sided bounds of the form τ^{-d/2}(π−arccos(I1+I2)+τ)^{-γ-d/2+1/2}(I2∨τ)^{-γ} exp(−arccos²(I1+I2)/(4τ)) uniformly for τ∈(0,1]. The proofs express the conic kernels through the Jacobi interval heat kernel via Lemmas 2.1 and 2.2 and then apply estimates due to Nowak, Sjögren, and Szarek. The paper also honestly records the obstructions for odd hyperboloid kernels and for the paraboloid settings. However, the decisive final comparison in Section 3 is asserted without proof and is not uniform, so the main claims are not established.

Significance. If the estimates were correct, they would be a valuable extension of the sharp Jacobi and spherical heat kernel bounds of NSS21 to double cones and hyperboloids, with explicit dependence on the boundary parameters. The initial reduction to the interval kernel is transparent, the parameter bookkeeping is mostly careful, and the paper correctly identifies the known obstructions for the paraboloid and the odd hyperboloid. The self-cited work [HK23] is not used as a load-bearing input. However, the central uniform comparison fails, and the stated theorems do not follow from the proof.

major comments (2)

[Section 3, proof of Theorem 1.6] The displayed comparison after Lemma 2.5, namely (I2/(π−arccos(I1+I2))+τ)^{−γ} ≃ (I2+τ)^{−γ} ≃ (I2∨τ)^{−γ}, is not uniform on V0^{d+1}. Fix a unit vector e1 and take t=s=1−ε, x=t e1, y=−s e1 with 0<ε≪1. Then I1=−(1−ε)^2, I2=2ε−ε^2, and I1+I2=−1+4ε−2ε^2, so π−arccos(I1+I2)=2√(2ε)+O(ε^{3/2}). For τ=ε^{3/4}, we have I2/(π−arccos(I1+I2)) ≈ √(ε/2) and I2∨τ = τ, so the ratio of (I2/(π−arccos(I1+I2))+τ)^{−γ} to (I2∨τ)^{−γ} is approximately ε^{γ/4}, which tends to 0 for every γ>0. Thus the two sides are not comparable, and the proof of Theorem 1.6 fails at this step. Corollaries 1.7, 1.17 and 1.21 inherit the gap.
[Section 3, proof of Theorem 1.12] The statements that 'π−arccos ξ(1,v) is comparable to a constant' and that 'π−arccos ξ(1,1) is comparable to a constant' are false. On V^{d+1}, take t=s=1, x=e1, y=−e1; then I1=−1 and I2=I3=0, so ξ(1,1)=−1 and π−arccos ξ(1,1)=0. In the near-boundary regime t=s=1−ε, x=(1−ε−δ)e1, y=−(1−ε−δ)e1 with δ=ε^{3/4}, one has I3≈2δ and π−arccos ξ(1,1)≈2√(ε+δ), so for τ=ε^{3/4} the factor (I3/(π−arccos ξ(1,1))+τ)^{−µ} is not comparable to (I3∨τ)^{−µ}. The same nonuniformity affects the replacement involving I2. Therefore the proof of Theorem 1.12 fails, and Corollary 1.13 is not established.

minor comments (4)

[Equation (1.10)] The weight notation in formula (1.10) omits the parameter μ: it should read P^O_n(w_{β,γ,μ}; ...) = ... P^E_n(w_{β+1,γ,μ}; ...).
[Table 1] The last two rows of Table 1 both label the domain ~V^{d+1}_0; one of them should presumably be the solid paraboloid ~V^{d+1}.
[Paragraph before Lemma 2.3] The text refers to 'the Jacobi heat kernel on B^d', but Lemma 2.3 concerns the Jacobi kernel on the interval [−1,1]; the wording should be corrected.
[Proof of Theorem 1.12] The opening line fixes '(x,y),(y,s)∈V^{d+1}' instead of '(x,t),(y,s)∈V^{d+1}'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proofs reduce the conic kernels to Xu's closed reproducing-kernel formulas and invoke independent NSS estimates; no fitted parameter or self-citation is load-bearing.

full rationale

The derivation chain is self-contained relative to published external results. Lemmas 2.1 and 2.2 express the even Jacobi heat kernels exactly as integrals of the classical Jacobi heat kernel G^{λ,λ}, using Xu's closed forms (1.2) and (1.9). Lemma 2.3 is the known sharp bound for G^{λ,λ} from NSS21, and Lemmas 2.4–2.5 are independent integral estimates from the same source; none of these are authored by the present paper, and none are asserted without external proof. The only self-citation, [HK23], appears in a list of prior sharp-estimate results and plays no role in the proofs. Theorem 1.6 and 1.12 are not obtained by fitting parameters to the target quantities; the factors (I2∨τ)^−γ etc. emerge from the integral estimates. There is a serious mathematical gap at the phrase 'After noticing that' in Section 3, where the comparability (I2/(π−arccosξ(1)) + τ)^−γ ≃ (I2∨τ)^−γ is asserted without proof and is in fact not uniform near antipodal boundary points; however, that is a correctness flaw in an intermediate estimate, not a circular reduction of the theorem to its own inputs. The claimed estimates are not equivalent by construction to any fitted input, and no load-bearing premise is justified only by self-citation. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on Xu's framework and on NSS21 estimates, which are external results. The main internal assumption is the unproved simplification of the boundary factor; this is where the derivation gap appears.

assumptions (4)

domain assumption Xu's reproducing kernel formulas (1.2), (1.9), (1.15), (1.19) and the associated diffusion operators are valid in the stated parameter ranges.
The paper builds on [Xu21] without reproving these constructions; if any formula or eigenvalue is wrong, the kernel identifications fail.
domain assumption The sharp Jacobi heat kernel estimate (Lemma 2.3) and integral estimates (Lemmas 2.4, 2.5) from [NSS21] hold in the stated uniformity.
These are cited as known and are used verbatim in the main proofs.
ad hoc to paper The comparison (I2/(pi - arccos(I1+I2)) + tau)^{-gamma} is comparable to (I2 or tau)^{-gamma}.
Never proved; used twice in the proofs of Theorems 1.6 and 1.12. It appears not to hold uniformly when tau lies between the two boundary scales for t and s near 1.
domain assumption Known uniform estimates give 'comparable to 1' for tau > 1.
Stated in Comment (d) without a citation; secondary to the small-time results.

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Cite this review

Pith. "Pith review of Sharp estimates for Jacobi heat kernels in double conic domains." pith.science (2026). https://pith.science/paper/NNSCMOXL

@misc{pith2026241115793,
  author       = {Pith},
  title        = {Pith review of: Sharp estimates for Jacobi heat kernels in double conic domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNSCMOXL}},
  note         = {Machine review of arXiv:2411.15793}
}

read the original abstract

We study the even and odd Jacobi heat kernels defined in the context of the multidimensional double cone and its surface, the multidimensional hyperboloid and its surface, and the multidimensional paraboloid and its surface. By integrating the framework of Jacobi polynomials on these domains, as analyzed by Xu, with contemporary methods developed by Nowak, Sj\"ogren, and Szarek for obtaining sharp estimates of the spherical heat kernel, we establish genuinely sharp estimates for the even and odd Jacobi heat kernel in the double conic settings, and for the even Jacobi heat kernel on hyperbolic settings. We also discuss the limitations encountered in extending these results to the remaining settings.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Heat kernel estimates for the Bessel differential operator in half-line

Bogus, K., Ma ecki, J. Heat kernel estimates for the Bessel differential operator in half-line. Math. Nachr. 289 (2016), 2097--2107

work page 2016

[2] [2]

Heat kernel bounds on hyperbolic space and Kleinian groups

Davies, E.B., Mandouvalos, N. Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. London Math. Soc. 57 (1988), 182--208

work page 1988

[3] [3]

Approximation theory and harmonic analysis on spheres and balls

Dai, D., Xu, Y. Approximation theory and harmonic analysis on spheres and balls. Springer Monographs in Mathematics, Springer, 2013

work page 2013

[4] [4]

Orthogonal Polynomials of Several Variables

Dunkl, C.F., Xu, Y. Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge, 2014

work page 2014

[5] [5]

Fourier-Bessel heat kernel estimates

Ma ecki, J., Serafin, G., \.Z\'orawik T. Fourier-Bessel heat kernel estimates. J. Math. Anal. Appl. 439 (2016), 91--102

work page 2016

[6] [6]

Dirichlet heat kernel for the laplacian in a ball

Ma ecki, J., Serafin, G. Dirichlet heat kernel for the laplacian in a ball. Potential Anal. 52 (2020), 545--563

work page 2020

[7] [7]

Sharp estimates of the Jacobi heat kernel

Nowak, A., Sj \"o gren, P. Sharp estimates of the Jacobi heat kernel. Studia Math. 218 (2013), 219--244

work page 2013

[8] [8]

Sharp estimates of the spherical heat kernel

Nowak, A., Sj \"o gren, P., Szarek, T.Z. Sharp estimates of the spherical heat kernel. J. Math. Pures Appl. 129 (2018), 23--33

work page 2018