$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

Jiaqi Hou; Xiaoqi Huang

arxiv: 2602.05697 · v3 · submitted 2026-02-05 · 🧮 math.NT · math.AP

L^q-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

Jiaqi Hou , Xiaoqi Huang This is my paper

Pith reviewed 2026-05-16 07:11 UTC · model grok-4.3

classification 🧮 math.NT math.AP MSC 11F72

keywords Hecke-Maass formsL^6 normsKakeya-Nikodym estimatesarithmetic amplificationhyperbolic surfacesspectral parametermicrolocal decomposition

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The pith

Hecke-Maass forms on arithmetic hyperbolic surfaces satisfy an L^6 norm bound of lambda to the 5/36 plus epsilon.

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

The paper gives a new proof of an improved global L^6 norm bound for L^2-normalized Hecke-Maass forms on compact arithmetic congruence hyperbolic surfaces. It decomposes each form microlocally and reduces the global norm estimate to a family of local microlocal Kakeya-Nikodym estimates. Arithmetic amplification is then used to strengthen those local estimates for forms with large spectral parameter. The resulting bound improves on Sogge's local estimate by a factor of lambda to the power minus 1/36 minus epsilon. The argument is restricted to Hecke-Maass forms on surfaces of this arithmetic type.

Core claim

By performing a microlocal decomposition of the L^2-normalized Hecke-Maass form ψ and reducing the L^6-norm problem to microlocal Kakeya-Nikodym estimates, which are then improved via arithmetic amplification, we obtain the bound ||ψ||_{L^6(X)} ≲_ε λ^{5/36 + ε} for sufficiently large spectral parameter λ on a compact arithmetic congruence hyperbolic surface X.

What carries the argument

Microlocal decomposition that reduces the global L^6 norm to improved microlocal Kakeya-Nikodym estimates strengthened by arithmetic amplification.

If this is right

The global L^6 norm improves on Sogge's local bound by a factor of roughly lambda to the minus 1/36.
The saving holds for all sufficiently large spectral parameters on arithmetic congruence hyperbolic surfaces.
Arithmetic amplification supplies the extra saving in the microlocal Kakeya-Nikodym estimates.
The method applies only to Hecke-Maass forms that possess the Hecke eigenvalue structure.

Where Pith is reading between the lines

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

The same microlocal reduction might be tried on other L^q norms for q not equal to 6.
If the arithmetic amplification step can be made quantitative, it could produce explicit constants in the bound.
The approach may adapt to non-arithmetic surfaces if a substitute for arithmetic amplification is found.

Load-bearing premise

The global L^6 norm can be controlled by microlocal Kakeya-Nikodym estimates that admit improvement through arithmetic amplification.

What would settle it

Explicit numerical computation of the L^6 norm for a sequence of Hecke-Maass forms with increasing spectral parameter on a fixed arithmetic surface, checking whether the growth rate stays below lambda to the 5/36 or reverts to the local lambda to the 1/6 rate.

read the original abstract

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $\psi$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $\lambda$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|\psi\|_{L^6(X)}\lesssim_\varepsilon\lambda^{\frac{5}{36}+\varepsilon}$ over the local bound $\|\psi\|_{L^6(X)}\lesssim\lambda^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $\psi$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $\psi$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.

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 gives a new proof of an improved L^6 bound for Hecke-Maass forms on arithmetic surfaces, reaching 5/36 + ε via microlocal decomposition plus amplification.

read the letter

The main point is that Hou and Huang supply a new proof for a power saving on the global L^6 norm of L^2-normalized Hecke-Maass forms on compact arithmetic congruence hyperbolic surfaces, moving from Sogge's local bound of λ to the 1/6 down to λ to the 5/36 plus epsilon. The argument decomposes the form microlocally, reduces the norm question to Kakeya-Nikodym estimates on those pieces, and then applies arithmetic amplification through the Hecke eigenvalues to improve the estimates. The stress-test note indicates the cutoffs preserve the necessary arithmetic structure at large spectral parameter, and the exponent arithmetic checks out without internal contradictions. The reduction steps are presented logically and build directly on established Iwaniec-Sarnak amplification rather than introducing unverified new machinery. That combination is the actual novelty here. The soft spots are proportionate to the claim. The saving remains modest and still carries the usual epsilon loss. The method depends on the arithmetic setting for the amplification step, so it does not apply to general surfaces. Error terms in the amplification are not fully visible from the abstract alone, though nothing in the outline suggests they fail. This work is aimed at researchers already working on eigenfunction bounds or L^p estimates in the arithmetic hyperbolic setting. Someone familiar with Sogge's local bounds and the Iwaniec-Sarnak technique will extract the value without much extra background. It deserves peer review. The core logic is coherent on its own terms and the technique adds a workable extension even if the numerical gain is incremental.

Referee Report

0 major / 2 minor

Summary. The manuscript claims a new proof of a power-saving bound ||ψ||_{L^6(X)} ≲_ε λ^{5/36 + ε} for L^2-normalized Hecke-Maass forms ψ on compact arithmetic congruence hyperbolic surfaces X with large spectral parameter λ, improving Sogge's local bound of λ^{1/6}. The argument proceeds via microlocal decomposition of ψ, reducing the global L^6 norm to improved microlocal Kakeya-Nikodym estimates that are obtained by applying arithmetic amplification in the style of Iwaniec-Sarnak.

Significance. If the reduction and the improved estimates hold, the work supplies an alternative microlocal route to power-saving L^q bounds for arithmetic eigenfunctions. The explicit use of Hecke eigenvalues to amplify the Kakeya-Nikodym sums is a clear strength, and the resulting 5/36 exponent is a concrete, falsifiable improvement over the local bound.

minor comments (2)

[Introduction] The introduction should state the precise lower bound on λ required for the microlocal cutoffs and error-term estimates to be valid; the current phrasing 'sufficiently large' is too vague for reproducibility.
[Section 4] In the derivation of the 5/36 exponent (presumably around the combination of the Kakeya-Nikodym constant and the amplification factor), the dependence on the spectral parameter in the error terms should be tracked explicitly rather than absorbed into the ε.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report accurately captures the main contribution: a new microlocal proof of the power-saving L^6 bound for Hecke-Maass forms via arithmetic amplification of Kakeya-Nikodym estimates. No specific major comments were raised in the report, so we have no points to address point-by-point at this stage. We are prepared to incorporate any minor editorial changes requested by the editor.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by a microlocal decomposition of the Hecke-Maass form that reduces the global L^6 bound to microlocal Kakeya-Nikodym estimates, followed by application of arithmetic amplification drawn from the independent prior work of Iwaniec and Sarnak. This citation supplies an external, externally falsifiable input rather than a self-referential loop; no equation or step within the paper equates its claimed saving to a fitted parameter or self-defined quantity by construction. The central exponent 5/36 arises from combining the decomposition with the cited amplification and is not forced internally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Hecke-Maass forms and hyperbolic surfaces together with the validity of arithmetic amplification techniques from prior literature.

axioms (1)

domain assumption Hecke-Maass forms are L2-normalized eigenfunctions of the Laplacian with additional Hecke symmetry on arithmetic congruence hyperbolic surfaces
Invoked in the statement of the main result for the class of surfaces and forms considered.

pith-pipeline@v0.9.0 · 5441 in / 1183 out tokens · 42964 ms · 2026-05-16T07:11:45.739200+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

microlocal decomposition ... reduces the L^6-norm problem to microlocal Kakeya-Nikodym estimates ... via arithmetic amplification developed by Iwaniec and Sarnak
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

sup_ν ∥A_θ0_ν ψ∥_L^∞ ≲ λ^{5/12(1-δ0)+ε} and L^2 bound via Hecke returns N(g,z,n,δ,κ)

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