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arxiv: 2503.01094 · v3 · submitted 2025-03-03 · 🧮 math.CA

Hardy's Theorem for the (k,frac{2}{n})-Fourier Transform

Hanen Jilani , Selma Negzaoui This is my paper

Pith reviewed 2026-05-23 02:14 UTC · model grok-4.3

classification 🧮 math.CA
keywords Hardy's theoremuncertainty principle(k, 2/n)-Fourier transformPhragmén-Lindelöf lemmaheat equationMiyachi theoremCowling-Price theorem
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The pith

The (k, 2/n)-Fourier transform obeys a Hardy-type uncertainty principle obtained by Gaussian comparison.

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

The paper establishes that Hardy's uncertainty principle extends to the (k, 2/n)-Fourier transform. The proof proceeds by comparing any admissible function and its transform to the Gaussian e^{-na|x|^{2/n}} and then invoking the Phragmén-Lindelöf lemma. A dynamical form of the principle is derived by studying the associated heat equation. The same comparison technique also yields Miyachi-type and Cowling-Price-type statements in mixed L^p-L^q norms.

Core claim

By comparing a function and its (k, 2/n)-Fourier transform to the Gaussian analogue e^{-na|x|^{2/n}}, the authors establish a Hardy-type uncertainty principle using the Phragmén-Lindelöf lemma. They further obtain a dynamical version via the heat equation and extend to Miyachi-type and Cowling-Price-type theorems in L^p-L^q settings.

What carries the argument

The (k, 2/n)-Fourier transform together with direct comparison to the Gaussian e^{-na|x|^{2/n}} and application of the Phragmén-Lindelöf lemma.

If this is right

  • Pointwise decay bounds on both a function and its (k, 2/n)-Fourier transform cannot hold simultaneously beyond the Gaussian threshold.
  • The uncertainty bound evolves in time according to the heat equation associated with the transform.
  • Analogous decay statements hold when the L^2 norms are replaced by L^p and L^q norms.
  • The same comparison argument produces Cowling-Price and Miyachi forms of the principle.

Where Pith is reading between the lines

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

  • The Gaussian-comparison method may transfer to other parameter-dependent Fourier transforms without requiring new complex-analysis tools.
  • The dynamical version suggests that uncertainty spreads at a rate governed by the heat kernel of the (k, 2/n) operator.
  • Similar extensions could be attempted for transforms indexed by different exponents or for operators on non-Euclidean domains.

Load-bearing premise

The Phragmén-Lindelöf lemma applies once the function and its transform have been bounded by the given Gaussian.

What would settle it

An explicit function f such that both f and its (k, 2/n)-Fourier transform decay slower than any multiple of the Gaussian e^{-na|x|^{2/n}} yet still satisfy the conclusion of the stated uncertainty principle would falsify the claim.

read the original abstract

By comparing a function and its $(k, \frac{2}{n})-$Fourier transform to a Gaussian analogue, $e^{-na|x|^\frac{2}{n}}$, we establish a Hardy-type uncertainty principle using Phragm\'en-Lindl\"of lemma. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy's theorem that illustrates the temporal evolution of the uncertainty principle. We also extend our results to $L^p-L^q$ versions, proving Miyachi-type and Cowling-Price-type theorems for the $(k,\frac{2}{n})$-Fourier transform.

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.

Referee Report

2 major / 2 minor

Summary. The paper establishes a Hardy-type uncertainty principle for the (k, 2/n)-Fourier transform by comparing a function f and its transform to the Gaussian e^{-na |x|^{2/n}} (and its counterpart) and invoking the Phragmén-Lindelöf lemma. It further derives a dynamical version of the theorem via the associated heat equation and extends the results to L^p-L^q settings, including Miyachi-type and Cowling-Price-type theorems.

Significance. If the central application of Phragmén-Lindelöf holds, the work would provide a parameterized extension of classical uncertainty principles with dynamical and L^p-L^q variants, which could be of interest in harmonic analysis on generalized Fourier transforms.

major comments (2)
  1. [Proof of the Hardy-type theorem] The proof of the main Hardy-type theorem (via comparison to e^{-na |x|^{2/n}} and Phragmén-Lindelöf) must explicitly construct a holomorphic branch of |z|^{2/n} in the relevant sector and verify that both f and its (k, 2/n)-transform admit analytic continuations compatible with the pointwise majorant; the abstract and skeptic note leave this verification unaddressed, which is load-bearing for the lemma invocation.
  2. [Dynamical version / heat equation section] For the dynamical version via the heat equation, the manuscript needs to confirm that the time-evolved function preserves the Gaussian comparison and the sectorial holomorphy required by Phragmén-Lindelöf at each t > 0; without this, the temporal evolution claim does not follow directly from the static case.
minor comments (2)
  1. [Introduction / preliminaries] Notation for the (k, 2/n)-Fourier transform should be defined with explicit kernel and domain before the first theorem statement.
  2. [L^p-L^q extensions] The abstract mentions extensions to Miyachi-type and Cowling-Price-type theorems but does not indicate whether the proofs reuse the same Phragmén-Lindelöf argument or require separate estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive suggestions. We agree that the proofs require additional explicit details on the holomorphic constructions and verifications to make the applications of the Phragmén-Lindelöf lemma fully rigorous. We will incorporate these clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Proof of the Hardy-type theorem] The proof of the main Hardy-type theorem (via comparison to e^{-na |x|^{2/n}} and Phragmén-Lindelöf) must explicitly construct a holomorphic branch of |z|^{2/n} in the relevant sector and verify that both f and its (k, 2/n)-transform admit analytic continuations compatible with the pointwise majorant; the abstract and skeptic note leave this verification unaddressed, which is load-bearing for the lemma invocation.

    Authors: We agree that the current manuscript does not spell out the holomorphic branch construction or the analytic continuation verifications. In the revised version we will add a dedicated subsection that: (i) defines the sector S = {z ∈ ℂ : |arg z| < π n / 2} (adjusted for the exponent 2/n), (ii) constructs the principal holomorphic branch of |z|^{2/n} via the logarithm on S, and (iii) verifies that both f and its (k, 2/n)-Fourier transform extend holomorphically to S while satisfying the pointwise majorant |F(z)| ≤ C exp(−a |z|^{2/n}) (and the analogous bound for the transform). This will render the invocation of Phragmén-Lindelöf fully justified. revision: yes

  2. Referee: [Dynamical version / heat equation section] For the dynamical version via the heat equation, the manuscript needs to confirm that the time-evolved function preserves the Gaussian comparison and the sectorial holomorphy required by Phragmén-Lindelöf at each t > 0; without this, the temporal evolution claim does not follow directly from the static case.

    Authors: We concur that the dynamical statement requires an explicit check at each t > 0. In the revision we will insert a paragraph showing that the solution u(·, t) of the associated heat equation satisfies the same Gaussian majorant with a time-dependent constant a(t) > 0, and that the sectorial holomorphy established for the static case carries over to u(·, t) for every fixed t > 0 (via the explicit representation or semigroup properties). This will allow direct application of the Hardy-type result at each time slice. revision: yes

Circularity Check

0 steps flagged

No circularity; proof applies external Phragmén-Lindelöf lemma to Gaussian comparison

full rationale

The derivation compares functions to the Gaussian e^{-na|x|^{2/n}} and invokes the Phragmén-Lindelöf lemma, an external result. No self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatz smuggling appear in the provided abstract or description. The central uncertainty principle is obtained by direct application of the lemma after the comparison, without reducing to the paper's own inputs by construction. This is the normal case of an independent proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on applicability of the Phragmén-Lindelöf lemma to the generalized setting after Gaussian comparison; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Phragmén-Lindelöf lemma applies to the functions and transforms after comparison to the Gaussian analogue
    Invoked directly to obtain the uncertainty principle from the comparison.

pith-pipeline@v0.9.0 · 5636 in / 1053 out tokens · 59619 ms · 2026-05-23T02:14:58.252926+00:00 · methodology

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Reference graph

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