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Improvement of a Hardy-Littlewood inequality and applications to the boundedness of analytic paraproducts on mixed norm spaces

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Pith's one-line read An improved Hardy-Littlewood inequality characterizes when analytic paraproducts are bounded between mixed-norm spaces.

desk verdict The paper sharpens the Hardy-Littlewood inequality with radius-independent constants and uses it to give necessary-and-sufficient conditions for boundedness of the three analytic paraproducts on the mixed-norm spaces A^{p,q}_ω, while settling one case of Luecking's Carleson problem. read the letter →

arxiv 2605.28080 v1 pith:IXMGS5S7 submitted 2026-05-27 math.CV math.CAmath.FA

Álvaro Miguel Moreno , José Ángel Peláez This is my paper · ORCID
classification math.CVmath.CAmath.FA
keywords Hardy-LittlewoodinequalityanalyticparaproductsmixednormspacesradialdoublingweightsCarlesonmeasuresboundedoperatorsHardy
topics The Measurement Problem
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

The paper sharpens the classical Hardy-Littlewood inequality that relates integral means of different orders for analytic functions at different radii. This sharpened form is applied to obtain exact conditions on a symbol g that make the three analytic paraproducts T_g, S_g, and M_g bounded operators between distinct mixed-norm spaces A^{p,q}_ω built from a radial doubling weight ω. The same improvement also settles a concrete case of Luecking's open Carleson-measure question. A reader would care because these paraproducts are fundamental operators whose mapping properties on weighted spaces often translate directly into growth or integrability conditions on the symbol.

What carries the argument

The sharpened Hardy-Littlewood inequality whose radius-independent constants convert integral estimates for the paraproducts into precise boundedness criteria on the spaces A^{p,q}_ω.

What would settle it

An explicit analytic function f together with radii r<ρ for which the improved inequality fails to hold with a uniform constant, or a symbol g making one of the paraproducts bounded on the mixed-norm spaces without satisfying the derived integral condition on g.

Watch

Extended reading notes

Core claim

We obtain an improvement of the Hardy-Littlewood inequality M_q(r,f) ≤ C(p,q) M_p(ρ,f) / (ρ-r)^{1/p-1/q} for 0≤r<ρ≤1 that holds with constants independent of r and ρ. This improvement is employed to characterize the symbols g∈H(D) such that the analytic paraproducts T_g f(z)=∫_0^z f(ζ)g'(ζ)dζ, S_g f(z)=∫_0^z f'(ζ)g(ζ)dζ and M_g f(z)=f(z)g(z) are bounded between two different mixed-norm spaces A^{p,q}_ω induced by a radial doubling weight ω. En route we solve a meaningful particular case of Luecking's open Carleson measure problem.

Load-bearing premise

The sharpened Hardy-Littlewood inequality holds with constants independent of the radii r and ρ and is strong enough to convert the integral estimates for the paraproducts into the precise boundedness criteria on A^{p,q}_ω.

Editorial extensions

If this is right

  • The boundedness of each of T_g, S_g, and M_g between distinct A^{p,q}_ω spaces is equivalent to an explicit integrability condition on the symbol g with respect to the weight ω.
  • A concrete case of Luecking's Carleson-measure problem admits a positive solution.
  • The same sharpened inequality yields boundedness criteria for the three paraproducts when the source and target spaces differ in both the p and q parameters.

Reading between the lines

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

  • The radius-independent sharpening may extend the reach of mean-value estimates to other radial weights or to non-doubling weights.
  • The solved Carleson-measure instance could serve as a test case for attacking the general Luecking problem via similar integral-mean techniques.
  • The method supplies a template for obtaining symbol criteria for other integral operators built from analytic functions on mixed-norm spaces.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper claims an improvement of the classical Hardy-Littlewood inequality M_q(r,f) ≤ C(p,q) M_p(ρ,f) / (ρ-r)^{1/p - 1/q} for analytic f, with the sharpened version having constants independent of 0 ≤ r < ρ ≤ 1. This improvement is inserted into integral estimates for the analytic paraproducts T_g, S_g and M_g to obtain necessary and sufficient conditions on g ∈ H(D) for boundedness between distinct mixed-norm spaces A^{p,q}_ω induced by radial doubling weights ω. En route, a special case of Luecking’s open Carleson-measure problem is solved.

Significance. If the claimed improvement holds with r,ρ-independent constants and is strong enough to convert the paraproduct integral estimates into exact boundedness criteria, the work supplies a useful sharpened tool for operator theory on weighted analytic spaces and resolves a concrete instance of an open Carleson-measure question. The combination of a pointwise inequality refinement with explicit operator characterizations on A^{p,q}_ω is a substantive contribution to the field.

major comments (2)
  1. [Abstract and §2] The abstract states that an improvement exists and suffices for the characterizations, yet the reader’s report notes that neither the precise statement of the sharpened inequality nor a proof sketch appears in the abstract; the full manuscript must therefore contain an explicit formulation (presumably in §2 or §3) together with the derivation showing independence of r and ρ. If that derivation is only sketched, the central claim remains load-bearing and requires a self-contained verification.
  2. [§4 (applications to paraproducts)] The weakest assumption identified is that the sharpened Hardy-Littlewood form converts the integral estimates for ||T_g f||_{A^{p,q}_ω}, ||S_g f|| and ||M_g f|| into precise conditions on g via a solved special case of Luecking’s problem. The manuscript must verify that the constants remain uniform under the doubling hypothesis on ω and that no hidden dependence on r,ρ re-enters when the inequality is integrated against ω.
minor comments (2)
  1. [Introduction] Notation for the mixed-norm spaces A^{p,q}_ω should be introduced once, with the precise definition of the integral ∫ M_p^q(r,g) ω(r) dr made explicit before the statements of the main theorems.
  2. [§3] The statement of the solved special case of Luecking’s Carleson-measure problem should be isolated as a separate theorem or proposition, with the precise measure condition written out.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The positive assessment of the contribution is appreciated. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §2] The abstract states that an improvement exists and suffices for the characterizations, yet the reader’s report notes that neither the precise statement of the sharpened inequality nor a proof sketch appears in the abstract; the full manuscript must therefore contain an explicit formulation (presumably in §2 or §3) together with the derivation showing independence of r and ρ. If that derivation is only sketched, the central claim remains load-bearing and requires a self-contained verification.

    Authors: The explicit statement of the sharpened inequality appears as Theorem 2.1, with a complete self-contained proof in Section 2 establishing r,ρ-independence of the constants. The derivation is fully detailed rather than sketched. We will revise the abstract to include a concise formulation of the improved inequality. revision: yes

  2. Referee: [§4 (applications to paraproducts)] The weakest assumption identified is that the sharpened Hardy-Littlewood form converts the integral estimates for ||T_g f||_{A^{p,q}_ω}, ||S_g f|| and ||M_g f|| into precise conditions on g via a solved special case of Luecking’s problem. The manuscript must verify that the constants remain uniform under the doubling hypothesis on ω and that no hidden dependence on r,ρ re-enters when the inequality is integrated against ω.

    Authors: The proofs of the characterizations in Section 4 (Theorems 4.1–4.3) explicitly verify uniformity of constants under the radial doubling condition on ω. The integration against ω is performed in detail, and the doubling property is used to ensure no r,ρ-dependence is reintroduced. The special case of Luecking’s problem is solved self-containedly in Section 3. We will add a clarifying remark on the uniformity if needed for emphasis. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper starts from the classical Hardy-Littlewood inequality and derives an improved pointwise estimate with r-independent constants; this sharpened form is inserted into the integral expressions for the three paraproducts to obtain equivalent conditions on the symbol g via a solved special case of Luecking’s Carleson-measure problem. No equation or characterization reduces by construction to a fitted parameter, self-definition, or self-citation chain; the weight-doubling hypothesis is used only for the standard doubling properties and the Carleson solution is presented as an auxiliary result rather than a tautological input. The derivation chain is therefore self-contained against the external classical inequality and Luecking’s open problem.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Review performed from abstract only; the ledger therefore records only the background facts explicitly invoked in the abstract. No free parameters or invented entities are visible. The listed axioms are the minimal domain assumptions required to state the spaces and operators.

assumptions (2)
  • standard math Analytic functions on the unit disk satisfy the subharmonicity and mean-value properties used to define M_p(r,f) and M_∞(r,f).
    Invoked in the first paragraph when the classical inequality is stated.
  • domain assumption The weight ω is radial and doubling, so that the mixed-norm spaces A^{p,q}_ω are well-defined Banach or quasi-Banach spaces.
    Explicitly required for the target spaces in the abstract.

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Pith. "Pith review of Improvement of a Hardy-Littlewood inequality and applications to the boundedness of analytic paraproducts on mixed norm spaces." pith.science (2026). https://pith.science/paper/IXMGS5S7

@misc{pith2026260528080,
  author       = {Pith},
  title        = {Pith review of: Improvement of a Hardy-Littlewood inequality and applications to the boundedness of analytic paraproducts on mixed norm spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXMGS5S7}},
  note         = {Machine review of arXiv:2605.28080}
}
abstract

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. For $0<p<\infty$ and $f\in\mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)=\int_0^{2\pi}|f(re^{i\theta})|^p \frac{d\theta}{2\pi}$ and $M_\infty(r,f) = \sup_{|z|=r}|f(z)|$. For $0<p<q\leq \infty$, Hardy and Littlewood proved the prevalent inequality $$M_q(r,f)\le C(p,q)\frac{M_p(\rho,f)}{(\rho-r)^{\frac{1}{p}-\frac{1}{q}}}$$ for $0\leq r<\rho\leq 1$ and $f\in\mathcal{H}(\mathbb{D})$. In this paper, we obtain an improvement of this well-known inequality which is employed to characterize the symbols $g\in\mathcal{H}(\mathbb{D})$ such that the analytic paraproducts $T_gf(z)=\int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)=\int_0^z f'(\zeta)g(\zeta)\,d\zeta$ and $M_gf(z)=f(z)g(z)$, are bounded between two different mixed-norm spaces $A^{p,q}_\omega=\{ g\in \mathcal{H}(\mathbb{D}): \int_0^1 M_p^q(r,g) \omega(r)\,dr<\infty\}$ induced by a radial doubling weight $\omega$. En route to the proof of these characterizations, we consider an open Carleson measure problem posed by Luecking and we solve it in a meaningful particular case.

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