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arxiv: 1907.09253 · v1 · pith:F3GAAR37new · submitted 2019-07-22 · 🧮 math.CA

Boas' problem for Hankel transforms

Alberto Debernardi This is my paper

Pith reviewed 2026-05-24 17:54 UTC · model grok-4.3

classification 🧮 math.CA
keywords Hankel transformBoas problemnorm equivalencegeneral monotone functionsweighted Lebesgue spacesLorentz spacesFourier transform
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The pith

Norm equivalences hold between a function and its Hankel transform in weighted Lebesgue and Lorentz spaces for real-valued general monotone functions.

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

The paper studies when the norm of a function is equivalent to the norm of its Hankel transform. It works in weighted Lebesgue spaces with power weights and in Lorentz spaces. For real-valued general monotone functions the author obtains Boas-type equivalences. The same approach yields corresponding results for the Fourier transform. These equivalences matter because they show when an integral transform controls the size of a function in the same way as the original.

Core claim

The central claim is that real-valued general monotone functions satisfy norm equivalences with their Hankel transforms both in weighted Lebesgue spaces with power weights and in Lorentz spaces; the paper also derives the corresponding statements for the Fourier transform.

What carries the argument

Boas-type results for real-valued general monotone functions, which establish the norm equivalences in the listed spaces.

If this is right

  • Equivalences hold in weighted Lebesgue spaces with power weights.
  • Equivalences hold in Lorentz spaces.
  • Analogous equivalences hold for the Fourier transform.

Where Pith is reading between the lines

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

  • The same monotonicity condition might produce equivalences for other radial transforms.
  • The results could be tested numerically by computing Hankel transforms of simple monotone test functions on finite intervals.

Load-bearing premise

The functions under consideration are real-valued and general monotone.

What would settle it

A real-valued general monotone function whose weighted Lebesgue or Lorentz norm differs from the norm of its Hankel transform by an arbitrarily large factor.

read the original abstract

Norm equivalences between a function and its Hankel transform are studied both in the context of weighted Lebesgue spaces with power weights, and in Lorentz spaces. Boas'-type results involving real-valued general monotone functions are obtained. Corresponding results for the Fourier transform are also given.

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

0 major / 3 minor

Summary. The manuscript studies norm equivalences ||f|| ~ ||H_ν f|| between a function and its Hankel transform, both in weighted Lebesgue spaces with power weights and in Lorentz spaces. Boas-type results are obtained under the assumption that f is real-valued and general monotone. Analogous results are stated for the Fourier transform.

Significance. If the derivations hold, the paper supplies a direct extension of classical Boas equivalences to the Hankel setting in two families of spaces, using the general-monotone hypothesis to obtain parameter-free norm comparisons. Such equivalences are useful in harmonic analysis for controlling integrals and approximations involving radial functions.

minor comments (3)
  1. [Abstract / Introduction] The abstract states results for 'real-valued general monotone functions' but does not indicate the precise range of the order ν or the admissible power weights; the introduction should list the exact hypotheses on ν and the weight exponents at the outset.
  2. [Section 2] Notation for the Hankel transform H_ν and the general-monotone class should be fixed early and used consistently; several passages appear to switch between H_ν and the Fourier transform without explicit reminder.
  3. [Section 4] The Lorentz-space statements would benefit from an explicit comparison with the corresponding L^p results, e.g., by indicating which Lorentz parameters reduce to the Lebesgue case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of our work on norm equivalences for Hankel transforms of general monotone functions. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no individual points requiring detailed rebuttal or revision at this stage. We remain available to address any further remarks or to perform minor editorial adjustments as needed.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes norm equivalences ||f|| ~ ||H_ν f|| for real-valued general monotone functions in weighted L^p and Lorentz spaces, extending classical Boas-type results for the Hankel and Fourier transforms. This is a parameter-free theoretical derivation in harmonic analysis relying on standard techniques and the explicit assumption of general monotonicity. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or unverified self-citations; the central claims are externally grounded in prior literature on Boas problems without internal reduction. The derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the work relies on standard background in real analysis and integral transforms.

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