Riesz Theorem and Riesz-Fej\'er inequality for weighted harmonic Bergman spaces with applications to M\"obius invariant spaces

Himadri Halder; Rohit Kumar

REVIEW 1 major objections 6 minor 1 cited by

Riesz Theorem and Riesz-Fej\'er inequality for weighted harmonic Bergman spaces with applications to M\"obius invariant spaces

T0 review · 1 major / 6 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read For K-quasiregular harmonic maps, membership of the real part in a weighted harmonic Bergman space forces the imaginary part into the same space, with a constant independent of the weight when p>1.

desk verdict Solid, expected extension of Riesz conjugation and Riesz–Fejér to weighted harmonic Bergman spaces under quasiregularity; the p≤1 case and the Q-space applications are the real new pieces, and the proofs check out. read the letter →

arxiv 2607.11795 v1 pith:LN5O3CSB submitted 2026-07-13 math.CV

Himadri Halder , Rohit Kumar This is my paper · ORCID

classification math.CV MSC 31A0530H2030C62

keywords RieszconjugatetheoremRiesz–FejérinequalityweightedharmonicBergmanspacesMöbiusinvariantquasiregularmappingsK-quasiregular

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 proves that if a harmonic K-quasiregular mapping has its real part in a weighted harmonic Bergman space a_α^p, then its imaginary part lies in the same space and the norms are comparable. When p>1 the comparison constant does not depend on the weight parameter α. The same circle of ideas yields Riesz–Fejér inequalities that bound the integral of |f|^p along a diameter by a multiple of the weighted area integral of |f|^p; the constant is improved for the Hilbert-space case p=2 by exploiting orthogonality of the monomials. Because the Möbius-invariant spaces Q(n,p,α) and their harmonic analogues embed continuously into weighted Bergman spaces, the same conjugate and diameter inequalities transfer immediately to those families.

What carries the argument

The harmonic conjugation operator T:u↦v on K-quasiregular maps, controlled first by the known Hardy-space Riesz theorem (for p>1) and then integrated against the weight (1-r^{2})^α r dr, or by local mean-value and quasiregular dilatation estimates (for 0<p≤1).

What would settle it

Construct a sequence of K-quasiregular harmonic maps whose real parts stay bounded in a_α^p while the imaginary parts blow up in a_α^p, or show that any comparison constant must grow with α when p>1.

Watch

Extended reading notes

Core claim

If f=u+iv is harmonic and K-quasiregular in the disk and u belongs to a_α^p (α>-1, 0<p<∞), then v also belongs to a_α^p and ||v|| ≤ C_{p,K,α} ||u||; moreover C is independent of α whenever 1<p<∞. Parallel Riesz–Fejér inequalities hold for the same spaces, with a sharper constant when p=2.

Load-bearing premise

For p greater than 1 the argument simply multiplies an already-known Hardy-space conjugate bound by the radial weight and integrates; if that Hardy bound fails or secretly depends on extra parameters, the weight-independence claim collapses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does exactly what the abstract claims: it fills the missing weighted-harmonic-Bergman slot between the known Hardy-space Riesz theorem for K-quasiregular maps and the unweighted Bergman version. Theorem 1.1 is the main statement; for p>1 the constant is independent of α, which is clean and useful. Theorems 1.2–1.3 give the corresponding Riesz–Fejér inequalities, with a genuine improvement at p=2 via orthogonality. The applications to Zhu’s Q(n,p,α) and the harmonic Qh spaces are then immediate once the continuous embeddings into weighted Bergman spaces are written down.

What is actually new is the weighted setting for harmonic (not just analytic) functions under the quasiregularity hypothesis, plus the explicit p≤1 argument. Case 1 of the proof of 1.1 simply integrates the Liu–Zhu Hardy-space bound against the radial weight; that is legitimate inheritance, not circularity, and the α-independence falls out for free. The real work is Case 2: local mean-value control (Theorem 3.1), the derivative bound at 0, Möbius change of variables, Tonelli, and the subharmonic comparison of Lemma 3.2. Those steps are standard but carefully written and do not lean on the Hardy black box. The Riesz–Fejér proofs follow the classical dilation-plus-Fubini pattern; the p=2 improvement is a nice use of the Hilbert-space structure.

Soft spots are minor. The p>1 constant inherits whatever dependence on p and K is already present in the Hardy-space literature; that is not a flaw of this paper. The constants themselves are not sharp, and the applications to Q/Qh are essentially corollaries rather than deep new characterizations. Citation pattern is appropriate—Zhu, Liu–Zhu, Das–Rasila, Peláez–Rättyä—and self-citation is limited to background.

This is for people who work on harmonic quasiregular mappings or Möbius-invariant spaces and need the weighted Bergman versions on the shelf. The math is elementary complex analysis that can be checked line-by-line. I would send it to a referee; it is a clean, self-contained advance in a specialized corner and deserves a journal slot after ordinary polishing.

Referee Report

1 major / 6 minor

Summary. The paper proves a Riesz conjugate theorem for weighted harmonic Bergman spaces a_α^p: if f=u+iv is harmonic K-quasiregular in the disk and u∈a_α^p (0<p<∞, α>−1), then v∈a_α^p with a quantitative bound; for 1<p<∞ the constant is independent of α. It also establishes Riesz–Fejér inequalities for a_α^p when 1<p<∞, with a sharper constant for p=2 obtained via orthogonality of the harmonic expansion. These results are applied to obtain conjugate and Riesz–Fejér statements for Zhu’s Möbius-invariant spaces Q(n,p,α) and the harmonic spaces Q_h(n,p,α) of Sun–Liu–Wang, in the range where those spaces coincide with (harmonic) Bloch spaces.

Significance. The work fills a natural gap between the known Riesz theorems for harmonic Hardy spaces of quasiregular maps (Liu–Zhu, Chen et al.) and the unweighted Bergman case (Das–Rasil a), and supplies the first Riesz–Fejér inequalities for harmonic Bergman spaces. The p≤1 half of Theorem 1.1 is the main analytic contribution (local mean-value estimates, automorphism changes of variable, and a subharmonic-type comparison). The p=2 improvement via Hilbert-space orthogonality is clean and useful. The applications to Q/Q_h are of interest once correctly proved, since those spaces unify Bloch, Besov and Q_s-type spaces. The arguments are classical and self-contained once the cited Hardy-space black box is granted; there are no free parameters or circular definitions.

major comments (1)

Theorem 2.1 (and its analytic counterpart): the proof claims that Q_h(n,p,α)⊂a_{α−np}^p together with Theorem 1.1 immediately yields v∈Q_h. This is incorrect: Theorem 1.1 only returns membership in the Bergman space, not in Q_h. In the stated range α>np−1 one has Q_h=B_h (and Q=B), so the result is true, but the argument must use the Bloch identification and the gradient comparison of Remark 2.1 (or an equivalent direct estimate on the Q-seminorm). The same gap appears for the analytic case via [29, Cor. 6]. Rewrite the proof of Theorem 2.1 so that the main line is the Bloch argument; the Bergman embedding alone does not close the claim.

minor comments (6)

Proof of Theorem 1.2, display after (3.13): the evaluation of ∫(1−r^{2})^α r dr is typeset ambiguously as “1 2 (1+α)”. Clarify that it equals 1/(2(1+α))(1−u^{2})^{α+1} so the factor 2π C_p (and the final π sec^p(π/(2p))) is transparent.
Case 2 of Theorem 1.1: the reduction “WLOG f(0)=0” should be justified in one line (subtract the constant; the conjugate of a real constant may be taken zero).
Lemma 3.1 / proof of Theorem 1.1: the radii ε=1/2, ε/3 and the disk D_{1/2}(0) are slightly inconsistent in the write-up; a single fixed radius (e.g. 1/2) throughout would improve readability.
Theorem 2.2: the right-hand side is written in terms of the Bergman norm ||f||_{a_{α−np}^p}. Since the embedding of Lemma 2.1 controls that norm by the Q-seminorm, it would be clearer to state the bound directly with ||f||_{Q_h} (or note the comparison).
Several references appear as arXiv preprints with future journal years (e.g. [6], [7], [9], [30]); update status if published versions exist before final submission.
Minor typos: “Resul ts”, “Applica tion”, “inv ariant”, “sp aces” in section headings; “doamin” (p. 12); “writting” (p. 14).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: weighted Riesz and Riesz–Fejér statements are ordinary analytic reductions of prior Hardy/Bergman bounds plus self-contained local estimates.

full rationale

The paper’s derivation chain is standard complex analysis and does not define its target norms or constants in terms of the claimed conclusions. For Theorem 1.1, Case 1 (p>1) simply integrates the already-published Hardy-space bound Mp(r,v)≤Cp,K Mp(r,u) of Liu–Zhu against the radial weight 2(1+α)(1-r^{2})α r dr; the resulting constant is therefore independent of α by construction and inherits only the known dependence on p and K. Case 2 (0<p≤1) proceeds from a local mean-value inequality (Theorem 3.1), a pointwise derivative bound under quasiregularity (Lemma 3.1), automorphism change-of-variable, and a subharmonic-type integral comparison (Lemma 3.2); none of these steps is self-definitional or fitted. Theorems 1.2–1.3 reduce the Riesz–Fejér claim to the known harmonic Hardy-space inequality of Melentijević–Božin (or Andreev’s L^{2} analytic bound) by dilation and Fubini, again without free parameters. Applications to Q(n,p,α) and Qh(n,p,α) rest on continuous embeddings into weighted Bergman spaces (Lemma 2.1) that are proved by direct evaluation at a=0 and the standard derivative characterization of Bergman spaces; the citations to Zhu and to Sun–Liu–Wang merely name the spaces being studied and do not force the conclusions. No uniqueness theorem is imported from the authors, no ansatz is smuggled via self-citation, and no empirical fit is renamed a prediction. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely inside classical complex analysis on the unit disk. No free parameters are fitted; the only numerical constants that appear are absolute or depend only on the structural parameters p, K, α. Background results (Riesz conjugation on Hardy spaces for quasiregular maps, derivative characterizations of Bergman spaces, Möbius invariance of Q-spaces) are taken from the literature and used as black boxes. No new physical or geometric entities are postulated.

assumptions (4)

domain assumption Riesz conjugate theorem for harmonic K-quasiregular maps in Hardy spaces h^p (1<p<∞): Mp(r,v) ≤ C_{p,K} Mp(r,u).
Imported from Liu–Zhu / Chen et al. and used as the starting point for the p>1 case of Theorem 1.1 (display (3.8)).
standard math Derivative characterization of weighted Bergman spaces: f ∈ A_β^p iff (1-|z|^{2})^n f^{(n)} ∈ L^p(dA_β).
Cited from Zhu’s book and used in Lemma 2.1 to embed Q(n,p,α) into A_{α-np}^p.
standard math Interior gradient estimate for harmonic functions and Fefferman–Stein-type pointwise bounds on disks of radius ≤1/3.
Used in Theorem 3.1 and Lemma 3.1 to control |h'(0)| by a local weighted average of |u|.
domain assumption K-quasiregularity implies |g'| ≤ k|h'| with k=(K-1)/(K+1)<1.
Definition of the class under study; appears throughout the estimates that relate |f| to |h'|.

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Cite this review

Pith. "Pith review of Riesz Theorem and Riesz-Fej\'er inequality for weighted harmonic Bergman spaces with applications to M\"obius invariant spaces." pith.science (2026). https://pith.science/paper/LN5O3CSB

@misc{pith2026260711795,
  author       = {Pith},
  title        = {Pith review of: Riesz Theorem and Riesz-Fej\'er inequality for weighted harmonic Bergman spaces with applications to M\"obius invariant spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LN5O3CSB}},
  note         = {Machine review of arXiv:2607.11795}
}

abstract

The aim of this paper is twofold. First, we establish a Riesz conjugate theorem for weighted harmonic Bergman spaces. More precisely, we prove that if $f=u+iv$ is a harmonic $K$-quasiregular mapping in $\mathbb{D}$ and the real part $u$ belongs to the weighted harmonic Bergman space $a_\alpha^p$, $0<p<\infty$, then the imaginary part $v$ also belongs to the same space, together with a quantitative norm estimate. Moreover, for $1<p<\infty$, the corresponding constant is shown to be independent of the weight parameter $\alpha$. Second, we establish Riesz--Fej\'er inequalities for weighted harmonic Bergman spaces for $1<p<\infty$. In the special case $p=2$, we further improve the corresponding constant by using the Hilbert space structure and orthogonality techniques. As applications of our main results, we establish Riesz conjugate theorems and Riesz--Fej\'er inequalities for the M\"obius invariant spaces $Q(n,p,\alpha)$ introduced by Zhu [Illinois J. Math. 51 (2007), pp. 977--1002] and their harmonic counterparts $Q_h(n,p,\alpha)$ introduced by Sun, Liu, and Wang [Potential Anal. 65 (2026), Article no. 12].

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gabriel's and Frazer's problems for weighted Bergman spaces and their applications
math.CV 2026-07 reject novelty 5.0 of 10

Claims weighted Bergman and Q-space analogues of Gabriel/Frazer inequalities for analytic and harmonic functions, with full p>0 in the harmonic case; internal gaps and factor errors undercut the stated results.

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Works this paper leans on

34 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [1]

V. V. Andreev , Fej\'er--Riesz type inequalities for Bergman spaces, Rend. Circ. Mat. Palermo 61 (2012), 385--392

2012

[2] [2]

Astala and P

K. Astala and P. Koskela , H^p -theory for quasiconformal mappings, Pure Appl. Math. Q. 7 (2011), 19--50

2011

[3] [3]

E. F. Beckenbach , On a theorem of Fej\'er and Riesz, J. Lond. Math. Soc. 13 (1938), 82--86

1938

[4] [4]

A. P. Calder\'on , On theorems of M. Riesz and Zygmund, Proc. Amer. Math. Soc. 1 (1950), 533--535

1950

[5] [5]

Chen and H

S. Chen and H. Hamada , On (Fej\'er--)Riesz type inequalities, Hardy--Littlewood type theorems and smooth moduli, Math. Z. 305 (2023), Article no. 64

2023

[6] [6]

S. Chen, M. Huang, X. Wang and J. Xiao , Sharp Riesz conjugate functions theorems for quasiregular mappings, arXiv:2310.15452 (2025)

arXiv 2025

[7] [7]

Chen and D

S. Chen and D. Kalaj , Conjugate type properties of harmonic (K,K') -quasiregular mappings, arXiv:2509.17578 (2025)

arXiv 2025

[8] [8]

Das and A

S. Das and A. S. Kaliraj , A Riesz-Fejér type inequality for harmonic functions, J. Math. Anal. Appl. 507 (2022), 125812

2022