Harmonic conjugates on Bergman spaces induced by doubling weights

Jos\'e \'Angel Pel\'aez; Jouni R\"atty\"a

arxiv: 1907.10563 · v1 · pith:R66HPLQLnew · submitted 2019-07-24 · 🧮 math.CV · math.CA· math.FA

Harmonic conjugates on Bergman spaces induced by doubling weights

Jos\'e \'Angel Pel\'aez , Jouni R\"atty\"a This is my paper

Pith reviewed 2026-05-24 16:16 UTC · model grok-4.3

classification 🧮 math.CV math.CAmath.FA MSC 30H20

keywords Bergman spacesradial weightsdoubling weightsharmonic conjugationnorm equivalenceanalytic functionsweighted spaces

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The pith

For radial weights in hat D but not check D, the Bergman space norm is sharply estimated by quantities depending on the real part of the function.

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

This paper examines weighted Bergman spaces A^p_ω where the radial weight ω is in the class hat D but not in check D. These weights satisfy a one-sided doubling condition that makes harmonic conjugation fail to map the space to itself when p is at most 1, according to classical theorems. The authors nonetheless derive sharp estimates showing that the norm of an analytic function in this space can be bounded using expressions that involve only the real part of the function. They further prove that for some subclasses of these weights, these real-part based quantities actually serve as equivalent norms on the space. This matters because it separates the issue of norm control from the closure property under conjugation.

Core claim

We establish sharp estimates for the norm of the analytic Bergman space A^p_ω, with ω∈ hat D minus check D and 0<p<∞, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

What carries the argument

The classes of radial weights hat D and check D defined by integral doubling and reverse doubling conditions on the integral of the weight, used to compare the full Bergman norm to real-part dependent quantities.

If this is right

The norm equivalence holds for all positive p, extending beyond the range where conjugation fails.
For certain radial weights in hat D, the real-part quantities define an equivalent norm on A^p_ω.
The estimates are sharp, meaning the constants cannot be improved in general.
These results apply to the operator theory of Bergman spaces with such weights.

Where Pith is reading between the lines

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

Similar real-part control might hold for other function spaces like Hardy spaces with analogous weights.
The distinction between the two weight classes could be used to classify when other operators preserve the space.
Testing the estimates numerically for specific weights like logarithmic ones could verify sharpness.
Connections to boundary behavior of harmonic functions in these weighted spaces may be worth exploring.

Load-bearing premise

The radial weight belongs to hat D but not to check D.

What would settle it

Constructing a counterexample function in A^p_ω for some ω in hat D minus check D where the norm is not bounded by the real-part quantity, or finding a weight where the equivalence fails for the claimed classes.

read the original abstract

A radial weight $\omega$ belongs to the class $\widehat{\mathcal{D}}$ if there exists $C=C(\omega)\ge 1$ such that $\int_r^1 \omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$ for all $0\le r<1$. Write $\omega\in\check{\mathcal{D}}$ if there exist constants $K=K(\omega)>1$ and $C=C(\omega)>1$ such that $\widehat{\omega}(r)\ge C\widehat{\omega}\left(1-\frac{1-r}{K}\right)$ for all $0\le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $\omega\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p\le 1$. In this paper we establish sharp estimates for the norm of the analytic Bergman space $A^p_\omega$, with $\omega\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p<\infty$, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

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 delivers sharp real-part norm controls for A^p_ω on weights in hat D minus check D, plus equivalence results on some radial subclasses.

read the letter

The main point is that the authors prove sharp estimates for the analytic Bergman norm in terms of real-part quantities when the radial weight sits in hat D but not check D, and they show those quantities give equivalent norms on certain subclasses. This fills a gap left by the classical Hardy-Littlewood and Shields-Williams results, which already showed the harmonic Bergman space fails to be closed under conjugation for p ≤ 1 on exactly these weights. The new estimates work for all 0 < p < ∞ and are presented as sharp for the analytic case. The work sits squarely on the authors' own recent definition of the two weight classes, so the contribution is an application rather than a wholesale redefinition. The estimates look consistent with the doubling conditions once those are fixed, and the distinction between hat D and check D is used cleanly without circular appeal to the missing reverse-doubling property. The paper is short and focused; it does not claim broad new operator theory but sticks to the norm comparison. One soft spot is that the sharpness is asserted without explicit constants or a separate sharpness example in the abstract, though the full proofs presumably supply that. Readers will also need the prior paper to understand the weight classes, which is a minor but real dependency. This is written for specialists already working on weighted Bergman spaces and radial-weight operator theory on the disk. A reader outside that narrow area will not get much, but inside it the claims are concrete enough to be worth checking. The paper deserves a serious referee because the setting is well-defined, the extension of the classical results is genuine, and the weight conditions are reproducible. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to establish sharp estimates for the norm of the analytic Bergman space A^p_ω (ω radial, ω ∈ hat D minus check D, 0 < p < ∞) in terms of quantities depending on the real part of the function. It further asserts that these real-part quantities yield equivalent norms on certain subclasses of radial weights. The work relies on the authors' prior definition of the hat D and check D classes and on classical results (Hardy-Littlewood, Shields-Williams) showing that the harmonic Bergman space is not closed under conjugation precisely when ω lies in this difference class for p ≤ 1.

Significance. If the claimed sharp estimates hold, the results would supply explicit real-part control on analytic Bergman norms in the doubling-but-not-reverse-doubling regime, extending the classical p ≤ 1 picture to all p < ∞ and furnishing tools for operator-theoretic questions on these spaces. The explicit separation of the two weight classes, already shown natural in the authors' earlier work, is a concrete strength.

minor comments (3)

Abstract, line 3: 'we have recently prove' should read 'we recently proved'.
The manuscript should include at least one concrete example of a weight in hat D minus check D together with the explicit constants appearing in the norm estimates, to make the sharpness claim verifiable.
Notation for the real-part quantities (presumably introduced in §2 or §3) should be compared directly with the classical Hardy-Littlewood maximal function or the Shields-Williams integral means so that the improvement is transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper supplies explicit integral definitions for both weight classes directly in the text and invokes classical Hardy-Littlewood/Shields-Williams results on harmonic conjugation to obtain the stated norm equivalences for ω in hat D minus check D. The single self-reference to prior work merely contextualizes why the classes are natural; it does not supply the norm estimates or any load-bearing uniqueness theorem. No equation reduces to a fitted parameter, no ansatz is smuggled, and the central claims remain independent of the authors' earlier results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It rests on standard integral properties of radial weights on the disk and on the definitions of the two doubling classes given in the authors' prior work.

axioms (1)

standard math Standard properties of Lebesgue integrals and radial weights on the unit disk
Invoked when defining membership in hat D and check D via the integral inequalities.

pith-pipeline@v0.9.0 · 5806 in / 1144 out tokens · 25687 ms · 2026-05-24T16:16:39.180920+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pavlovi´ c and J

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J. A. Pel´ aez and J. R¨ atty¨ a, Bergman projection induc ed by radial weight, https://arxiv.org/abs/1902.09837, preprint (submitted )

work page arXiv 1902
[12]

A. L. Shields and D. L. Williams, Bounded projections an d the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), no. 1, 3–25

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Siskakis, Weighted integrals of analytic functions , Acta Sci

A. Siskakis, Weighted integrals of analytic functions , Acta Sci. Math. (Szeged) 66 (2000), no. 3-4, 651–664. Departamento de An ´alisis Matem ´atico, Universidad de M ´alaga, Campus de Teatinos, 29071 M´alaga, Spain. Phone number: 0034952131911 E-mail address : japelaez@uma.es University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland E-mail addr...

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[1] [1]

D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maxi mal function characterization of the class H p, Trans. Amer. Math. Soc. 157 (1971), 137–153

work page 1971

[2] [2]

Duren, Theory of H p Spaces, Academic Press, New York-London 1970

P. Duren, Theory of H p Spaces, Academic Press, New York-London 1970

work page 1970

[3] [3]

Garnett, Bounded analytic functions, Academic Press , New York, 1981

J. Garnett, Bounded analytic functions, Academic Press , New York, 1981

work page 1981

[4] [4]

G. H. Hardy and J. E. Littlewood, Some properties of conju gate functions, J. Reine. Angew. Math. 167 (1932), 405–423

work page 1932

[5] [5]

D. H. Luecking, A new proof of an inequality of Littlewood and Paley, Proc. Amer. Math. Soc. 103 (1988), no. 3, 887–893

work page 1988

[6] [6]

Pavlovi´ c, On harmonic conjugates with exponential m ean growth, Czechoslovak Math

M. Pavlovi´ c, On harmonic conjugates with exponential m ean growth, Czechoslovak Math. J. 49 (124) (1999), no. 4, 733–742

work page 1999

[7] [7]

Pavlovi´ c and J

M. Pavlovi´ c and J. A. Pel´ aez, An equivalence for weighted integrals of an analytic function and its deriv- ative, Math. Nachr. 281 (2008), no. 11, 1612–1623. 16 JOS ´E ´ANGEL PEL ´AEZ AND JOUNI R ¨ATTY ¨A

work page 2008

[8] [8]

J. A. Pel´ aez, Small weighted Bergman spaces, Proceedin gs of the summer school in complex and harmonic analysis, and related topics, (2016)

work page 2016

[9] [9]

J. A. Pel´ aez and J. R¨ atty¨ a, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014), no. 1066

work page 2014

[10] [10]

J. A. Pel´ aez and J. R¨ atty¨ a, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1-2, 205–239

work page 2015

[11] [11]

J. A. Pel´ aez and J. R¨ atty¨ a, Bergman projection induc ed by radial weight, https://arxiv.org/abs/1902.09837, preprint (submitted )

work page arXiv 1902

[12] [12]

A. L. Shields and D. L. Williams, Bounded projections an d the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), no. 1, 3–25

work page 1982

[13] [13]

Siskakis, Weighted integrals of analytic functions , Acta Sci

A. Siskakis, Weighted integrals of analytic functions , Acta Sci. Math. (Szeged) 66 (2000), no. 3-4, 651–664. Departamento de An ´alisis Matem ´atico, Universidad de M ´alaga, Campus de Teatinos, 29071 M´alaga, Spain. Phone number: 0034952131911 E-mail address : japelaez@uma.es University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland E-mail addr...

work page 2000