arxiv: 2605.09306 · v1 · submitted 2026-05-10 · 🧮 math.FA · math.OA· math.SP

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Weyl asymptotic formulas in the nilpotent Lie group setting

Dmitriy Zanin, Edward McDonald, Fedor Sukochev, Shiqi Liu

Pith reviewed 2026-05-12 02:15 UTC · model grok-4.3

classification 🧮 math.FA math.OAmath.SP

keywords Weyl asymptoticshypoelliptic operatorsgraded Lie groupsnilpotent Lie groupsspectral asymptoticssingular value estimatesanisotropic homogeneity

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The pith

Negative fractional powers of hypoelliptic operators on graded Lie groups obey a Weyl spectral asymptotic.

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

The paper derives a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. These operators have principal symbols that are homogeneous with respect to a non-isotropic dilation structure. The work shows how to extend such formulas from the constant-coefficient setting to operators whose coefficients vary smoothly. A sympathetic reader cares because the leading term of the asymptotic encodes the volume of the unit ball in the symbol space and therefore determines the distribution of eigenvalues or singular values for these operators.

Core claim

We derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols. Our methods allow extension from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is adaptation of singular-value perturbation arguments to the setting of nilpotent Lie groups, using a decomposition of graded Lie groups into homogeneous layers.

What carries the argument

Adaptation of singular-value perturbation arguments to nilpotent Lie groups equipped with a graded decomposition that respects the anisotropic homogeneity of the principal symbols.

If this is right

The eigenvalue or singular-value distribution of negative fractional powers is governed by the volume of the unit ball in the principal-symbol space.
The asymptotic extends to hypoelliptic operators whose coefficients are no longer constant.
A corresponding integration formula holds for traces of these operators on graded Lie groups.

Where Pith is reading between the lines

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

The same volume-based leading term may be used to define a noncommutative integral on the algebra generated by the operator.
The result suggests that spectral asymptotics can be obtained on other nilpotent groups once a suitable homogeneous decomposition is identified.
Numerical verification on low-dimensional examples such as the Heisenberg group would test the error-term control.

Load-bearing premise

The singular-value estimates that hold for constant-coefficient operators continue to hold for variable-coefficient operators on graded Lie groups without introducing new error terms that change the leading asymptotic.

What would settle it

An explicit computation of the eigenvalue counting function for a model hypoelliptic operator such as a sub-Laplacian on the three-dimensional Heisenberg group, compared against the volume predicted by the symbol.

read the original abstract

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.

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 adapts Birman-Solomyak perturbation to get Weyl asymptotics for negative fractional powers of variable-coefficient hypoelliptic operators on graded nilpotent groups, but the non-commutative error control needs checking.

read the letter

The main point is that this work supplies the first asymptotic formula for negative fractional powers of hypoelliptic operators with smoothly varying coefficients on graded nilpotent Lie groups by adapting the singular-value perturbation technique from Birman-Solomyak 1977, using the Folland-Stein decomposition for the graded structure. It also yields a version of Connes' integration formula in this setting. That is genuinely new and organizes eigenvalue counting for a class of operators that appear in sub-Riemannian geometry. The authors correctly note that the anisotropic homogeneity of the principal symbol makes the extension non-routine, and they frame the problem cleanly in terms of existing harmonic analysis on homogeneous groups. The paper does a reasonable job extending constant-coefficient Euclidean results without adding extraneous machinery. The soft spot is the adaptation step itself. The stress-test concern about commutator cross terms from the Baker-Campbell-Hausdorff formula is worth taking seriously: when the symbol is only anisotropically homogeneous, conjugating or perturbing the operator can produce remainder contributions that might sit at the same order as the leading Weyl term, especially for fractional powers. The abstract gives no explicit error estimates or verification that the leading coefficient survives unchanged, so the claim rests on whether the full proofs bound these terms tightly. If they do, the result holds; if the bounds are loose, the asymptotic may require extra assumptions. This paper is for people working in spectral theory on Lie groups and noncommutative geometry. A reader who already knows Birman-Solomyak and Folland-Stein will see the value in the extension. It deserves a serious referee because the result is new, the methods are grounded in prior literature, and the potential flaw is local rather than fatal to the whole argument. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper derives spectral asymptotic formulas for negative fractional powers of hypoelliptic operators on graded nilpotent Lie groups. It adapts singular-value perturbation arguments from Birman-Solomyak (1977) to this non-commutative setting via a Folland-Stein style decomposition of the group, extends the results from constant-coefficient to variable-coefficient operators, and obtains a version of Connes' integration formula as a consequence.

Significance. If the adaptation is carried through with controlled error terms, the work would usefully extend classical Weyl asymptotics to anisotropic symbols on graded groups and provide a concrete link to Connes' formula in the hypoelliptic case. The manuscript appropriately credits the foundational references and focuses on a technically natural generalization.

major comments (2)

The central technical step is the adaptation of the Birman-Solomyak singular-value perturbation argument. The manuscript must supply an explicit remainder estimate showing that commutators generated by the Baker-Campbell-Hausdorff formula on the graded group remain of strictly lower order than the leading Weyl coefficient when the principal symbol is only anisotropically homogeneous. Without this estimate the leading-term claim is not yet justified.
The passage from the eigenvalue asymptotic to the stated version of Connes' integration formula requires a precise statement of the trace-class remainder and the precise range of fractional powers for which the formula holds; these details are not fully spelled out in the current argument.

minor comments (2)

Abstract, line 3: 'the decomposing of graded Lie groups' should read 'the decomposition of graded Lie groups'.
Notation for the homogeneous degrees and the anisotropic symbol classes should be introduced once and used consistently; several ad-hoc symbols appear without prior definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: The central technical step is the adaptation of the Birman-Solomyak singular-value perturbation argument. The manuscript must supply an explicit remainder estimate showing that commutators generated by the Baker-Campbell-Hausdorff formula on the graded group remain of strictly lower order than the leading Weyl coefficient when the principal symbol is only anisotropically homogeneous. Without this estimate the leading-term claim is not yet justified.

Authors: We agree that an explicit remainder estimate is required to fully justify the leading term. The manuscript adapts the Birman-Solomyak argument in Section 3 using the Folland-Stein decomposition of the graded group, but the BCH commutator estimates are currently only indicated rather than stated as a separate lemma. In the revision we will add a precise lemma establishing that these commutators are of strictly lower anisotropic order than the principal symbol, thereby confirming that they do not affect the leading Weyl coefficient. revision: yes
Referee: The passage from the eigenvalue asymptotic to the stated version of Connes' integration formula requires a precise statement of the trace-class remainder and the precise range of fractional powers for which the formula holds; these details are not fully spelled out in the current argument.

Authors: We acknowledge that the derivation of the Connes-type formula needs greater precision. The asymptotic holds for negative fractional powers, but the manuscript does not isolate the trace-class remainder or specify the exact interval for the exponent. We will insert a dedicated corollary that states the formula for fractional powers -α with 0 < α < 1 (consistent with the hypoellipticity assumptions) and explicitly records that the remainder operator is trace-class. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation adapts external Birman-Solomyak 1977 perturbation arguments and Folland-Stein decomposition to graded groups

full rationale

The paper derives the Weyl-type asymptotic for negative fractional powers by adapting the singular-value perturbation technique of Birman-Solomyak (1977) to hypoelliptic operators with anisotropically homogeneous symbols on graded nilpotent Lie groups, using the Folland-Stein homogeneous decomposition. This is an extension from constant-coefficient to variable-coefficient cases and is presented as generalizing external prior results rather than reducing the leading coefficient or remainder to a self-defined fit, a renamed input, or a self-citation chain. No equations or steps in the provided abstract or description collapse the claimed asymptotic back onto the paper's own fitted quantities or prior self-referential theorems by construction. The central claim therefore remains independent of the inputs it starts from.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a suitable decomposition of graded Lie groups (inspired by Folland-Stein) and on the validity of singular-value perturbation estimates when transferred to the nilpotent setting; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Hypoelliptic operators on graded Lie groups admit anisotropically homogeneous principal symbols that allow a well-defined notion of negative fractional powers.
Invoked in the first sentence of the abstract as the setting in which the asymptotics are derived.
ad hoc to paper The singular-value perturbation arguments of Birman and Solomyak extend to the nilpotent Lie group setting without destroying the leading asymptotic term.
Stated as the principal technique; this is the load-bearing transfer step.

pith-pipeline@v0.9.0 · 5456 in / 1570 out tokens · 38051 ms · 2026-05-12T02:15:20.250685+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.

Foundation/AbsoluteFloorClosure.lean; Foundation/AlexanderDuality.lean reality_from_one_distinction unclear
Theorem 1.4... lim k→∞ k^{γ/d_hom} μ(k, M_f (P+1)^{-γ/m}) = (1/Γ(d_hom/m +1) ∫_G |f(g)|^{d_hom/γ} τ(exp(-P_top_g)) dg )^{γ/d_hom}

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