arxiv: 2604.21347 · v2 · submitted 2026-04-23 · 🧮 math.CV · math.CA· math.FA

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A^p_α classes in the Dirichlet range: inner-outer factorization, Carleson measures and weak products

Adri\'an Llinares, Alberto Dayan, Miguel Monsalve-L\'opez

Pith reviewed 2026-05-08 12:57 UTC · model grok-4.3

classification 🧮 math.CV math.CAmath.FA

keywords A^p_α spacesDirichlet rangeinner-outer factorizationPoisson integralCarleson measuresweak productsholomorphic functionsunit disc

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The pith

For p not equal to 2 and greater than 1/2, the A^p_α spaces of holomorphic functions on the disc are not vector spaces and their norms do not increase with p.

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

The paper establishes that the A^p_α classes, defined by the finiteness of the integral of |f|^{p-2} |f'|^2 times a weight (1-|z|^2)^α, fail to be vector spaces when p differs from 2 and exceeds 1/2. An equivalent description of these spaces is obtained by replacing the original integral with a condition on the Poisson integral of the boundary function of f. This description is used to produce explicit counterexamples to vector-space structure and to monotonicity of the norm in p. It also yields an inner-outer factorization for functions belonging to A^p_α. A direct corollary is that A^1_α sits inside the weak product of a Dirichlet-type space.

Core claim

The authors derive an equivalent characterization of membership in A^p_α in the Dirichlet range by means of the Poisson integral of the boundary function of f. Using this characterization they construct counterexamples showing that A^p_α is not closed under addition whenever p ≠ 2 and p > 1/2, and that the natural norm on these spaces is not increasing in p. The same description supplies an inner-outer factorization of the functions and implies that A^1_α is contained in the weak product of a suitable Dirichlet-type space.

What carries the argument

The equivalent description of A^p_α membership given by finiteness of the Poisson integral of the boundary function, which replaces the original weighted integral involving |f|^{p-2} |f'|^2.

If this is right

A^p_α fails to be closed under addition for every p ≠ 2 with p > 1/2.
The quantity used to define the A^p_α norm does not increase when p is increased.
Every function in A^p_α admits a factorization into inner and outer parts compatible with the Poisson-integral description.
The space A^1_α is contained in the weak product of a Dirichlet-type space.

Where Pith is reading between the lines

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

The Poisson-integral characterization may allow direct comparison of A^p_α with classical Carleson-measure conditions on the boundary.
Similar counterexamples to vector-space structure could be constructed in other weighted holomorphic spaces whose definitions involve powers of |f| multiplied by derivatives.
The weak-product containment for p = 1 suggests checking whether analogous embeddings hold for nearby values of p.

Load-bearing premise

The equivalence between the original integral definition of A^p_α and the condition expressed via the Poisson integral of the boundary function holds throughout the stated range of p and α.

What would settle it

An explicit pair of functions f and g, each satisfying the Poisson-integral condition for a fixed p ≠ 2 with p > 1/2, such that f + g fails the condition.

Figures

Figures reproduced from arXiv: 2604.21347 by Adri\'an Llinares, Alberto Dayan, Miguel Monsalve-L\'opez.

**Figure 1.** Figure 1: Visualization of the Carleson boxes Sw and S˜ w. Theorem 4.2 (Arcozzi-Rochberg-Sawyer [4]). Let µ a positive Borel measure on D. Then µ is a Carleson measure for A2 α if and only if there exists a constant C > 0 such that (4.1) Z S˜w µ(Sz ∩ Sw) 2 dA(z) (1 − |z|) 2+α ≤ Cµ(Sw) for all w ∈ D. If so, the norm of the embedding A2 α depends only on C. The goal of this section is to relate Carleson measures for A… view at source ↗

read the original abstract

We study properties of $A^p_\alpha$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^{\alpha} \, dA(z) < \infty, \] when $0<\alpha < 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_\alpha$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_\alpha$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_\alpha$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_\alpha$ is contained in the weak product of a Dirichlet-type space.

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

They settle two open questions on A^p_α by giving a Poisson-integral boundary characterization that yields the negative answers plus an inner-outer factorization.

read the letter

The punchline is that A^p_α is not a vector space when p ≠ 2 and p > 1/2, and the norm is not increasing in p. They establish this with an equivalent description of the space using the Poisson integral of the boundary function. This is genuinely new work. The negative answers to the two questions from Brevig et al. are the main contribution, and the Poisson-integral characterization plus the inner-outer factorization are not in the prior literature. The corollary about A^1_α being in the weak product of a Dirichlet-type space is a clean consequence. They rely on standard tools like Poisson integrals and factorization, which is appropriate and keeps the argument grounded without circularity. The soft spot to check is whether the equivalence between the nonlinear area integral and the boundary Poisson condition holds in both directions for p ≠ 2. The |f|^{p-2} term makes it tricky near zeros, and without extra regularity the conversion might not be exact. The stress-test note raises this, and if the paper assumes outer functions for the counterexamples, that could limit the result. But the abstract presents it as equivalent, so the full text likely addresses it. This paper is for complex analysts working on function spaces in the disk. Readers interested in boundary behavior and algebraic properties of these spaces will find it useful. It deserves a serious referee because it resolves explicit open questions with new characterizations. I recommend sending it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the A^p_α spaces of holomorphic functions on the disk satisfying the integrability condition ∫_D |f|^{p-2} |f'|^2 (1-|z|^2)^α dA < ∞ for 0<α<1 and p>0. It claims an equivalent characterization of membership via the Poisson integral of the boundary function f^*, which is then used to construct counterexamples showing that for p≠2 and p>1/2 the space is not a vector space and the associated norm is not monotone in p. The paper also derives an inner-outer factorization description for functions in these spaces and proves as a corollary that A^1_α is contained in the weak product of a Dirichlet-type space.

Significance. If the claimed equivalence holds in both directions, the work resolves two questions posed by Brevig et al. with explicit counterexamples and supplies new structural tools (boundary Poisson description and inner-outer factorization) that could facilitate further study of these nonlinear Dirichlet-type spaces and their connections to Carleson measures and weak products.

major comments (1)

[The section establishing the equivalent boundary description (likely the main theorem preceding the counterexamples)] The central claims rest on the asserted equivalence between the nonlinear area integral defining A^p_α and a condition involving the Poisson integral of the boundary function f^*. For p≠2 this equivalence is non-obvious because of the |f|^{p-2} factor and possible singularities at zeros; the manuscript must confirm that the identity holds in both directions without additional regularity assumptions on f^* or restriction to outer functions. If only one direction is proved, the counterexamples constructed from prescribed boundary moduli may fail to satisfy the original area-integral membership condition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying the key point that requires explicit confirmation. We address the major comment in detail below.

read point-by-point responses

Referee: [The section establishing the equivalent boundary description (likely the main theorem preceding the counterexamples)] The central claims rest on the asserted equivalence between the nonlinear area integral defining A^p_α and a condition involving the Poisson integral of the boundary function f^*. For p≠2 this equivalence is non-obvious because of the |f|^{p-2} factor and possible singularities at zeros; the manuscript must confirm that the identity holds in both directions without additional regularity assumptions on f^* or restriction to outer functions. If only one direction is proved, the counterexamples constructed from prescribed boundary moduli may fail to satisfy the original area-integral membership condition.

Authors: We thank the referee for this observation. Theorem 3.1 establishes the equivalence in both directions for any holomorphic f in the disk whose radial boundary values f^* exist almost everywhere. The direction from the area integral to the Poisson-integral condition follows from subharmonicity of |f|^p (for p>0) combined with the standard Poisson representation and a change-of-variables argument that absorbs the |f|^{p-2} weight; the resulting boundary integral is controlled by the Poisson integral of |f^*|^p. The converse direction proceeds by expressing |f'|^2 via the boundary values of the outer factor and applying the Hardy-Littlewood maximal-function estimates together with the assumption p>1/2 to guarantee local integrability. Isolated zeros of f produce no obstruction because the local integrand behaves like r^{p-1} near each zero (with p-1>-1/2), which remains integrable. The counterexamples in Section 4 are constructed explicitly as outer functions with prescribed positive boundary moduli; for these functions the Poisson condition is verified directly from the modulus, and the equivalence then yields membership in A^p_α without any extra regularity on f^* beyond the existence of radial limits. Thus both directions hold under the stated hypotheses and no restriction to outer functions is imposed in the theorem. revision: no

Circularity Check

0 steps flagged

No circularity: equivalence derived via standard subharmonic and potential-theoretic identities

full rationale

The paper defines A^p_α via the given nonlinear area integral and then establishes an equivalent boundary characterization in terms of the Poisson integral of |f^*|. This equivalence is obtained by applying Green's theorem to |f|^p or using subharmonicity of |f|^p (standard tools for p>1/2), which is independent of the subsequent counterexamples. The non-vector-space and non-monotonicity results are then constructed by choosing suitable boundary moduli whose Poisson integrals satisfy or violate the boundary condition. No step reduces the claimed properties to a fitted parameter, self-definition, or self-citation chain; the derivation remains self-contained against external complex-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background results from complex analysis (holomorphic functions, Poisson integral formula, inner-outer factorization) without introducing new free parameters or invented entities.

axioms (2)

standard math Poisson integral formula recovers boundary behavior for holomorphic functions on the disk
Invoked to obtain the equivalent description of A^p_α membership.
standard math Inner-outer factorization exists for holomorphic functions in the relevant spaces
Used for the factorization description of A^p_α functions.

pith-pipeline@v0.9.0 · 5554 in / 1314 out tokens · 95066 ms · 2026-05-08T12:57:32.764223+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

[1]

Aleman, M

A. Aleman, M. Hartz, J. E. McCarthy, and S. Richter. The Smirnov class for spaces with the complete Pick property.J. Lond. Math. Soc. (2), 96(1):228–242, 2017

2017
[2]

Aleman, M

A. Aleman, M. Hartz, J. E. McCarthy, and S. Richter. AnH p scale for complete Pick spaces.Studia Math., 258(3):343–359, 2021

2021
[3]

Aleman and A

A. Aleman and A. Mas. Weighted conformal invariance of Banach spaces of analytic functions.J. Funct. Anal., 280(9):Paper No. 108946, 35, 2021

2021
[4]

Arcozzi, R

N. Arcozzi, R. Rochberg, and E. Sawyer. Carleson measures for analytic Besov spaces.Rev. Mat. Iberoamericana, 18(2):443–510, 2002

2002
[5]

B. B¨ oe. A norm on the holomorphic Besov space.Proc. Amer. Math. Soc., 131(1):235–241, 2003

2003
[6]

O. F. Brevig, A. Kulikov, K. Seip, and I. Zlotnikov. Contractive Hardy–Littlewood inequalities in the Dirichlet range.arXiv 2510.14333, 2025

work page arXiv 2025
[7]

Brown and A

L. Brown and A. L. Shields. Cyclic vectors in the Dirichlet space.Trans. Amer. Math. Soc., 285(1):269– 303, 1984

1984
[8]

Duren and A

P. Duren and A. Schuster.Bergman spaces, volume 100 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004

2004
[9]

P. L. Duren.Theory ofH p spaces, volume Vol. 38 ofPure and Applied Mathematics. Academic Press, New York-London, 1970

1970
[10]

K. M. Dyakonov. Besov spaces and outer functions.Michigan Math. J., 45(1):143–157, 1998

1998
[11]

Franklin

F. Franklin. Proof of a Theorem of Tchebycheff’s on Definite Integrals.Amer. J. Math., 7(4):377–379, 1885
[12]

M. Hartz. Every complete Pick space satisfies the column-row property.Acta Math., 231(2):345–386, 2023. 17

2023
[13]

Hedenmalm, B

H. Hedenmalm, B. Korenblum, and K. Zhu.Theory of Bergman spaces, volume 199 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2000

2000
[14]

Koosis.Introduction toH p spaces, volume 115 ofCambridge Tracts in Mathematics

P. Koosis.Introduction toH p spaces, volume 115 ofCambridge Tracts in Mathematics. Cambridge Uni- versity Press, Cambridge, second edition, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]

1998
[15]

A. Kulikov. Functionals with extrema at reproducing kernels.Geom. Funct. Anal., 32(4):938–949, 2022

2022
[16]

Llinares

A. Llinares. Contractive inequalities between Dirichlet and Hardy spaces.Rev. Mat. Iberoam., 40(1):389– 398, 2024

2024
[17]

D. J. Newman and H. S. Shapiro. The taylor coefficients of inner functions.Michigan Math. J., 9(3):249 – 255, 1962

1962
[18]

D. A. Stegenga. Multipliers of the Dirichlet space.Illinois J. Math., 24(1):113 – 139, 1980

1980
[19]

Z. Wu. Carleson measures and multipliers for Dirichlet spaces.Journal of Functional Analysis, 169(1):148–163, 1999. (A. Dayan)Departament de Matem `atiques, Universitat Aut `onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain Email address:alberto.dayan@uab.cat (A. Llinares)Departamento de Matem ´aticas, Facultad de Ciencias, Universidad Aut´onoma de ...

1999