Note on real and imaginary parts of harmonic quasiregular mappings

Antti Rasila; Suman Das

REVIEW 3 major objections 4 minor 16 references

Note on real and imaginary parts of harmonic quasiregular mappings

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For harmonic quasiregular mappings, the real and imaginary parts share the same growth order and boundary smoothness.

desk verdict New results for 0<p<1 and the boundary Hölder transfer are real; Theorem 1's proof has a repairable gap in applying the converse of Hardy–Littlewood Theorem C. read the letter →

arxiv 2506.04618 v1 pith:QU3GLJST submitted 2025-06-05 math.CV

Suman Das , Antti Rasila This is my paper

classification math.CV MSC 31A0530H1030C62

keywords HardyspacesHarmonicquasiregularmappingsRiesztheoremConjugatefunctionsHöldercontinuityIntegralmeansQuasiconformal

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 shows that in a harmonic $K$-quasiregular mapping $f = u+iv$ in the unit disk, the real part $u$ and the imaginary part $v$ have the same order of growth of their integral means for every $0 < p \le \infty$, and the same Hölder smoothness on the boundary circle for every exponent $0 < \alpha < 1$. This extends the classical Riesz theorem for conjugate functions of analytic mappings, which is known to fail for general harmonic functions, to the quasiregular setting for the previously missing ranges $0

What carries the argument

The central device is the pair $f = h+g$ and $F = h+\overline{g}$, so that $\operatorname{Re} F = u$; quasiregularity enters as a uniform bound $|\omega| \le k<1$ on the dilatation $\omega = g'/h'$, which makes $M_p(r,F')$ comparable to $M_p(r,h')$ and $M_p(r,g')$. The Hardy–Littlewood theorem (Theorem C) is the bridge: a growth bound of order $\beta$ on the integral means of the real part of an analytic function forces a bound of order $\beta+1$ on the means of its derivative, and conversely for $\beta>0$. For the boundary result, Theorem E (Hölder continuity of $f$ on the circle iff $|f'(\rho e^{i\theta})| = O((1-\rho)^{\alpha-1})$) is applied after the Poisson integral shows that $u \in \Lambda_\alpha$ forces $F'$ to have exactly that growth.

What would settle it

To settle the $0<p<1$ case, check whether the converse of the Hardy–Littlewood theorem holds in the form used: does $M_p(r,\operatorname{Re} h) = O((1-r)^{-\beta})$ imply $M_p(r,|h'|) = O((1-r)^{-(\beta+1)})$ for an analytic $h$; exhibiting an analytic $h$ for which this fails would collapse the proof of Theorem 1, or a direct computation of $M_p(r,u)$ and $M_p(r,v)$ for a harmonic quasiregular mapping with chosen boundary data could reveal a rate mismatch that falsifies the theorem itself.

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Extended reading notes

Core claim

Writing $f = h + g$ with $g(0)=0$ and forming $F = h + \overline{g}$, the real part of $F$ equals $u$. Because $f$ is $K$-quasiregular, the complex dilatation $\omega = g'/h'$ satisfies $|\omega| \le k = (K-1)/(K+1) < 1$, so $|F'| = |(1+\omega)h'|$ controls $|h'|$ and $|g'|$ up to the factor $1-k$. The Hardy–Littlewood theorems then transfer an assumed growth bound on $M_p(r,u)$ first to $M_p(r,F')$, then to $M_p(r,h)$ and $M_p(r,g)$, and finally to $M_p(r,f)$ and $M_p(r,v)$, giving Theorem 1. The same transfer, combined with the Poisson integral representation of $F$ and a Hardy–Littlewood fractional-integrals criterion for Hölder continuity, yields Theorem 3: a Hölder exponent $\alpha \in (0,1)$ of the boundary values of $u$ is inherited by $v$, independently of $K$.

Load-bearing premise

The argument leans on the Hardy–Littlewood assertion that, for an analytic function, a growth bound on the means of its real part forces the same growth bound (with the exponent raised by one) on the means of its derivative, and that the converse holds for $\beta>0$; for $0<p<1$, the converse must be valid for the modulus rather than only for the real part, and if it is not, the proof of Theorem 1 in that range needs an additional step.

Editorial extensions

If this is right

The Riesz-type comparison of real and imaginary parts, previously known for $1<p<\infty$, now holds for all $0<p\le\infty$ when the harmonic map is quasiregular.
The boundary values of $u$ and $v$ share the same Hölder exponent $\alpha$ for every $0<\alpha<1$, with constants that do not depend on the quasiregularity constant $K$.
The endpoint $\alpha=1$ (Lipschitz) is genuinely exceptional: even the analytic case can produce an unbounded derivative of $v$ at a point, so no such transfer holds there.
For $u\in h^p$ with $0<p<1$, the imaginary part satisfies the logarithmic estimate $M_p(r,v) = O((\log 1/(1-r))^{1/p})$, showing the failure of Riesz's theorem in this range is quantitative rather than absolute.

Reading between the lines

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

A possible next step that goes beyond the paper is to ask whether the $O$-statements for $0<p<1$ can be sharpened to a constant-times inequality $M_p(r,v) \le C(K,p)M_p(r,u)$; the paper proves only the order.
Because the proof is inherently planar, it would not carry over to spatial quasiregular mappings without a new mechanism; that would be a separate project.
At the boundary, the failure at $\alpha=1$ suggests testing whether a $\log$-corrected (Zygmund-type) estimate holds exactly at Lipschitz regularity.
A concrete empirical check would be to compute the optimal constants in the growth transfer for small $p$ and compare with the known optimal Riesz constants when $K=1$.

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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 genuinely new material here is the order-of-growth comparison for harmonic quasiregular mappings in the range 0

The main soft spot is in the proof of Theorem 1. The authors get M_p(r,h')=O((1-r)^{-(\beta+1)}) and then say "the converse part of Theorem C shows" M_p(r,h)=O((1-r)^{-\beta}). As stated, the converse of Theorem C only recovers growth of the real part of an analytic function, not of its modulus. For p\ge1 the missing step is harmless, and even for p<1 it is repairable: apply the converse to get M_p(r,\Re h)=O((1-r)^{-\beta}), then invoke Theorem B (which the paper itself states) to get the same for \Im h, and use the quasi-norm inequality to sum. Alternatively one can integrate M_p(\rho,h') directly. But the manuscript does neither, so the proof of the central theorem is incomplete as written. This is a genuine gap, though a small one.

Two smaller issues: the representation f=h+g is missing a conjugate bar (it should be f=h+\bar g), and the statement of Theorem C's converse is imprecise enough to invite exactly the misstep above. The authors also rely on Lemma A from their own earlier paper in Theorem 2; that is fine since the lemma is a published independent result, and the application is legitimate (local univalence follows from quasiregularity).

Despite the gap, I am fairly confident the theorems are true, and the fixes are routine. The paper deserves a serious referee and could be accepted after minor revision. If you work on harmonic quasiregular mappings or Hardy-space growth, this is worth reading and citing. Otherwise it is a competent subfield note, not essential reading.

Referee Report

3 major / 4 minor

Summary. The paper studies harmonic K-quasiregular mappings f = u + iv in the unit disk and compares the growth and boundary regularity of the real and imaginary parts. Theorem 1 states that if M_p(r,u)=O((1-r)^{-\beta}) for some 0<p\le\infty and \beta>0, then M_p(r,v)=O((1-r)^{-\beta}). Theorem 2 gives a logarithmic bound for M_p(r,v) when u belongs to the harmonic Hardy space h^p, 0<p<1. Theorem 3 states that if the boundary values of u are H\"older continuous of order \alpha\in(0,1), then the boundary values of v have the same H\"older order. The proofs pass from u to the analytic function F=h+g whose real part is u, use quasiregularity to control h' and g', and then apply Hardy--Littlewood theorems to transfer growth or smoothness back to f and hence to v.

Significance. If the proofs are completed as indicated, the paper makes a clean contribution to the theory of harmonic quasiregular mappings: it extends the Riesz-type results of Liu--Zhu, Chen--Huang, Kalaj, and others from the range 1<p<\infty to every p>0, including p=1 and p=\infty, and it adds a new boundary H\"older-regularity transfer that is independent of the quasiregularity constant K. The proof strategy is economical, relying on standard Hardy--Littlewood theorems and on Lemma A from the authors' earlier work, and the paper is honest about limitations, explicitly noting that the H\"older result fails for \alpha=1. The main technical interest lies in the 0<p<1 case, where the usual subharmonicity and Minkowski arguments are unavailable; I found no circularity or parameter-fitting. The issues I raise below are local and repairable rather than fatal.

major comments (3)

[Section 1.1 and the proofs of Theorems 1--3] The harmonic representation is stated as f=h+g with h and g analytic. Taken literally, this would make every harmonic f analytic, and the subsequent quasiregularity condition \omega=g'/h' would not be the appropriate condition for a non-analytic harmonic map. The correct representation is f=h+\bar{g}; correspondingly, the proofs should define F=h+g, whose real part equals u=Re f. This notational error appears throughout the manuscript and must be corrected in a revision.
[Section 2, proof of Theorem 1] The step "the converse part of Theorem C shows M_p(r,h)=O((1-r)^{-\beta})=M_p(r,g)" is not justified by Theorem C as stated in Section 1.2. The stated converse is: M_p(r,f')=O((1-r)^{-(\beta+1)}) implies M_p(r,Re f)=O((1-r)^{-\beta}) for an analytic function f. It does not, by itself, give the corresponding growth for the modulus M_p(r,h). For 1\le p\le\infty this conclusion follows from Minkowski's integral inequality applied to the radial integral, and for 0<p<1 it follows from the subadditivity inequality |\int |^p\le \int |\,|^p (or by applying the stated converse to h and to -ih and then combining the estimates). This extra argument is not supplied, so the proof of the central theorem is incomplete as written.
[Section 2, proof of Theorem 2] Lemma A is invoked for the functions f_r(z)=f(rz), but the hypotheses of Lemma A are not checked. The proof should state that K-quasiregularity with |\omega|\le k<1 implies local univalence (and that the normalization f(0)=0 is harmless by a constant shift), so that f_r indeed satisfies the lemma. As written, this is a gap in the proof, although a routine one.

minor comments (4)

[Section 1.2] There are small typos: "They proved" should be "they proved", and "necessaily" should be "necessarily".
[Section 2, proof of Theorem 2] In the displayed estimate after applying Lemma A, the identity \|f_r\|_p^p = M_p^p(r,f) is implicit; it would help the reader if this were written explicitly.
[Section 2, Theorem 3] The assumption "u is continuous in D" should be phrased as "u is continuous in the closed unit disk", since the statement concerns boundary values u(e^{i\theta}).
[Example 1] The computation of the conjugate boundary function v(e^{i\theta})\sim\theta\log|\theta| would benefit from a citation or a short derivation, although the example is standard.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the proofs reduce to classical Hardy-Littlewood theorems and an independent prior lemma.

full rationale

The paper's derivation chain contains no circular step. Theorem 1 and Theorem 2 are proved via classical Hardy-Littlewood results (Theorems B and C), the quasiregularity condition, and Lemma A from Das-Kaliraj [5]; Theorem 3 uses a standard Poisson-integral estimate and Hardy-Littlewood Theorem E. Lemma A is a self-citation and is load-bearing in Theorem 2, but it is an independent published result whose hypotheses (locally univalent harmonic mappings with f(0)=0) do not include the present paper's conclusions, so under the rules it counts as independent support rather than circularity. No parameter is fitted, no entity is defined in terms of the target conclusion, and no ansatz is smuggled in via citation. The reader-flagged issue in Theorem 1, namely applying the converse of Theorem C to M_p(r,|h|) rather than to the real part of h for 0<p<1, is a possible correctness gap in a classical argument, not an instance of the paper assuming its conclusion; it is repairable and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest entirely on quoted theorems from the literature plus the quasiregularity definition. There are no fitted parameters or new postulated objects.

assumptions (5)

standard math Hardy-Littlewood Theorem C: M_p(r,u)=O((1-r)^-beta) implies M_p(r,F')=O((1-r)^-(beta+1)) for 0<p<=infinity, beta>=0, with converse for beta>0.
Invoked in proofs of Theorems 1 and 2 to transfer growth of u to F' and back to h and g. Quoted from [7] and not proved in this paper.
standard math Lemma A from Das-Kaliraj [5]: for locally univalent harmonic f=h+\bar{g}, ||f||_p^p <= C integral_0^1 (1-r)^(p-1) M_p^p(r,h') dr for 0<p<1.
Used in Theorem 2 to bound M_p(r,f) from M_p(r,h'). It is a published theorem from prior work by one of the authors and is not derived in this paper.
standard math Hardy-Littlewood Theorem E: analytic f extends continuously to the boundary with f(e^{i theta}) in Lambda_alpha iff |f'(r e^{i theta})|=O((1-r)^(alpha-1)), 0<alpha<1.
Used at the end of Theorem 3 to convert derivative growth of h and g into boundary Hölder continuity.
domain assumption Poisson integral representation of analytic F with Re F=u for continuous boundary data u.
Used in Theorem 3; requires u continuous on the closed disk, as assumed.
domain assumption A K-quasiregular harmonic map has representation f=h+\bar{g} with |g'| <= k|h'|, k=(K-1)/(K+1), and is locally univalent.
This is the definitional setting of the paper; it underlies the inequalities |F'| >= (1-k)|h'| and |g'| <= k|h'| used in all three theorems.

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Cite this review

Pith. "Pith review of Note on real and imaginary parts of harmonic quasiregular mappings." pith.science (2026). https://pith.science/paper/QU3GLJST

@misc{pith2026250604618,
  author       = {Pith},
  title        = {Pith review of: Note on real and imaginary parts of harmonic quasiregular mappings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QU3GLJST}},
  note         = {Machine review of arXiv:2506.04618}
}

abstract

If $f=u+iv$ is analytic in the unit disk $\mathbb{D}$, it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if $f$ is a (complex-valued) harmonic function. However, we prove that the same principle holds if we assume, in addition, that $f$ is $K$-quasiregular in $\mathbb{D}$. The case $0<p<1$ is particularly interesting, and is an extension of the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Further, we proceed to show that the real and imaginary parts of a harmonic quasiregular mapping have the same degree of smoothness on the boundary.

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Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

Astala and P

K. Astala and P. Koskela. H p-theory for quasiconformal mappings. Pure Appl. Math. Q. , 7(1):19–50, 2011

work page 2011
[2]

Sharp Riesz conjugate functions theorems for quasiregular mappings

S. Chen and M. Huang. Riesz type theorems for κ-pluriharmonic mappings, invariant harmonic quasiregular mappings and harmonic quasiregular mappings.arXiv preprint arXiv:2310.15452, 2023

work page Pith review arXiv 2023
[3]

S. Das, J. Huang, and A. Rasila. Hardy spaces of harmonic quasiconformal mappings and Baernstein’s theorem. arXiv preprint arXiv:2505.05028 , 2025

work page arXiv 2025
[4]

S. Das, J. Huang, and A. Rasila. Zygmund’s theorem for harmonic quasiregular mappings. Complex Anal. Oper. Theory , 19:Article no. 91, 13 pp., 2025

work page 2025
[5]

Das and A

S. Das and A. Sairam Kaliraj. Integral mean estimates for univalent and locally univalent harmonic mappings. Canad. Math. Bull. , 67(3):655–669, 2024

work page 2024
[6]

P. L. Duren. Theory of H p spaces. Pure and Applied Mathematics, Vol. 38. Academic Press, New York-London, 1970

work page 1970
[7]

G. H. Hardy and J. E. Littlewood. Some properties of conjugate functions. J. Reine Angew. Math., 167:405–423, 1932

work page 1932
[8]

G. H. Hardy and J. E. Littlewood. Some properties of fractional integrals. II. Math. Z. , 34(1):403–439, 1932

work page 1932

Show all 16 references

[9]

Hariri, R

P. Hariri, R. Kl´ en, and M. Vuorinen.Conformally invariant metrics and quasiconformal map- pings. Springer Monographs in Mathematics. Springer, Cham, 2020

2020
[10]

D. Kalaj. Riesz and Kolmogorov inequality for harmonic quasiregular mappings. J. Math. Anal. Appl., 542(1):Paper No. 128767, 15 pp., 2025

2025
[11]

D. Kalaj. Zygmund theorem for harmonic quasiregular mappings. Anal. Math. Phys., 15(2):Pa- per No. 41, 13 pp., 2025

2025
[12]

Li and S

P. Li and S. Ponnusamy. On the coefficients estimate ofK-quasiconformal harmonic mappings. arXiv preprint arXiv:2504.08284 , 2025

2025
[13]

Liu and J.-F

J. Liu and J.-F. Zhu. Riesz conjugate functions theorem for harmonic quasiconformal map- pings. Adv. Math., 434:Paper No. 109321, 27, 2023

2023
[14]

S. K. Pichorides. On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. , 44:165–179, 1972

1972
[15]

S. Rickman. Quasiregular mappings, volume 26 of Ergebnisse der Mathematik und ihrer Gren- zgebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, 1993

1993
[16]

Wang, X.-Y

Z.-G. Wang, X.-Y. Wang, A. Rasila, and J.-L. Qiu. On a problem of Pavlovi´ c involving harmonic quasiconformal mappings. arXiv preprint arXiv:2405.19852 , 2024. Suman Das Department of Mathematics with Computer Science, Guangdong Technion - Is- rael Institute of Technology, Sh...

2024

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Astala and P

K. Astala and P. Koskela. H p-theory for quasiconformal mappings. Pure Appl. Math. Q. , 7(1):19–50, 2011

work page 2011

[2] [2]

Sharp Riesz conjugate functions theorems for quasiregular mappings

S. Chen and M. Huang. Riesz type theorems for κ-pluriharmonic mappings, invariant harmonic quasiregular mappings and harmonic quasiregular mappings.arXiv preprint arXiv:2310.15452, 2023

work page Pith review arXiv 2023

[3] [3]

S. Das, J. Huang, and A. Rasila. Hardy spaces of harmonic quasiconformal mappings and Baernstein’s theorem. arXiv preprint arXiv:2505.05028 , 2025

work page arXiv 2025

[4] [4]

S. Das, J. Huang, and A. Rasila. Zygmund’s theorem for harmonic quasiregular mappings. Complex Anal. Oper. Theory , 19:Article no. 91, 13 pp., 2025

work page 2025

[5] [5]

Das and A

S. Das and A. Sairam Kaliraj. Integral mean estimates for univalent and locally univalent harmonic mappings. Canad. Math. Bull. , 67(3):655–669, 2024

work page 2024

[6] [6]

P. L. Duren. Theory of H p spaces. Pure and Applied Mathematics, Vol. 38. Academic Press, New York-London, 1970

work page 1970

[7] [7]

G. H. Hardy and J. E. Littlewood. Some properties of conjugate functions. J. Reine Angew. Math., 167:405–423, 1932

work page 1932

[8] [8]

G. H. Hardy and J. E. Littlewood. Some properties of fractional integrals. II. Math. Z. , 34(1):403–439, 1932

work page 1932