Extremal functions and zero sets for the Dirichlet space

Alexander Borichev (I2M); Karim Kellay (IMB); Omar El-Fallah

REVIEW 3 major objections 5 minor 22 references

Extremal functions and zero sets for the Dirichlet space

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single kernel sum decides which point sets are Dirichlet zero sets

desk verdict Real candidate for the first necessary-and-sufficient zero-set criterion for the Dirichlet space, but Theorem 3.1 is false as stated for sequences containing 0 and the key cited proposition is left unstated. read the letter →

arxiv 2608.14103 v1 pith:7Q4CJEZC submitted 2026-08-14 math.CA math.CVmath.FA

Alexander Borichev (I2M) , Omar El-Fallah , Karim Kellay (IMB) This is my paper · ORCID

classification math.CAmath.CVmath.FA MSC 46E2231A0531A1531A2047B32

keywords DirichletspacezerosetsextremalfunctionsreproducingkernelsharmonicspacesShapiro–Shieldsconditioninvariantsubspaces

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 a complete characterization of zero sets for the classical Dirichlet space: a sequence of points in the unit disk is a zero set if and only if a certain series, built recursively from extremal functions and diagonal values of reproducing kernels, converges. The construction tracks how much room a function vanishing at the first $n$ points leaves at the next point; the classical Shapiro–Shields condition is exactly the version of this series in which every intermediate measure is replaced by arc measure. Zero sets of the Dirichlet space are a long-studied topic, and a necessary-and-sufficient test was missing; this paper supplies one, together with geometric sufficient conditions for zero sets and zero divisors. A reader should care because the criterion turns an existence question about analytic functions into a computable convergence question.

What carries the argument

The load-bearing object is the diagonal reproducing-kernel sum attached to a sequence of measures generated by extremal functions. For a finite zero sequence $Z_n$, let $M_n$ be the subspace of functions vanishing on $Z_n$, and let $\phi_n$ be its extremal function, the unit vector in $M_n\ominus zM_n$; its boundary modulus defines the next measure $d\mu_n=|\phi_n|^2dm$. The kernel value $k^{\mu_{n-1}}_{z_n}(z_n)$ measures the squared norm of evaluation at $z_n$ in the intermediate space $D(\mu_{n-1})$, so $1/k^{\mu_{n-1}}_{z_n}(z_n)$ is the cost of forcing a zero at $z_n$ after the earlier zeros are already imposed. Formula (3) expresses the single-point extremal function $\phi_{a,\mu}$ in terms of the reproducing kernel, which converts the product representation of $\phi_n$ into the numerical criterion.

What would settle it

Evaluate the series (4) on a sequence already known by independent arguments to be a zero set that fails the classical sufficient condition, such as the examples discussed in [10], [11], [15]: the sum must converge. Evaluate it on a sequence known not to be a zero set: the sum must diverge. A single mismatch in either direction would refute Theorem 3.1.

Watch

Extended reading notes

Core claim

Let $Z=(z_n)_{n\ge 1}$ be a sequence in the unit disk. The central claim is Theorem 3.1: $Z$ is a zero set for the Dirichlet space if and only if $\sum_{n\ge 1} 1/k^{\mu_{n-1}}_{z_n}(z_n)<\infty$, where $\mu_0=m$ is normalized arc measure and, recursively, $d\mu_n=|\phi_n|^2dm$, with $\phi_n$ the extremal function of the invariant subspace $\{f\in\mathcal{D}: f(z_1)=\cdots=f(z_n)=0\}$. The proof factors each extremal function as a product of single-point extremal functions, whose values at $0$ are $\sqrt{1-1/k^{\mu_{j-1}}_{z_j}(z_j)}$; convergence of the series is equivalent to the product of these factors not collapsing to zero. Under the same condition the extremal functions converge weakly to a nonzero function vanishing on $Z$. The classical Shapiro–Shields sufficient condition appears as the special case in which all intermediate measures $\mu_{n-1}$ are replaced by arc measure $m$.

Load-bearing premise

The necessity direction rests on a cited weak-convergence statement about extremal functions for decreasing invariant subspaces, and the proof also assumes that $0$ is not among the zeros; if either premise fails, the characterization does not follow.

Editorial extensions

If this is right

Zero-set membership for a sequence $Z$ is decided by convergence of the recursively defined series (4); no separate construction of a function is needed for the positive direction.
The classical Shapiro–Shields theorem is a direct special case, recovered by setting every $\mu_{n-1}=m$ in (4), so the new criterion is strictly sharper.
The sufficient conditions in Theorems 4.1 and 4.3 turn the criterion into geometric tests: a Blaschke sequence lying in small regions attached to a known zero set is itself a zero set when the counting function of those regions is small.
For separated zero sequences the method yields zero divisors: multiplicities can be prescribed with $p_n$ growing like $\log^M(1/(1-|z_n|))$ while still admitting a nonzero function with those vanishing orders.

Reading between the lines

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

A numerical version of the criterion is within reach: formula (3) and the diagonal kernel estimates of [5] allow recursive approximation of $\phi_n$ and of the terms $1/k^{\mu_{n-1}}_{z_n}(z_n)$, so explicit sequences could be tested on a computer and certificates of zero-set status produced.
The same mechanism should give necessary-and-sufficient criteria for the wider class of spaces mentioned in Remark 3.3; at the Hardy-space endpoint the series should collapse to the Blaschke condition.
The criterion depends on the order of the points through the measures $\mu_{n-1}$; testing whether the sum is invariant under permutations of $Z$ would clarify whether there is a genuinely order-independent capacity behind zero-set membership.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

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

Two things to know before you read this. The main theorem is a genuine candidate for the first necessary-and-sufficient condition for Dirichlet zero sets, and the framework (building measures from extremal functions of finite prefixes) is new and natural. But Theorem 3.1 is false as stated for sequences containing 0, and the proof of the nontrivial direction leans on a proposition from the authors' earlier paper that is not stated.

The good: Lemma 2.2 gives the product formula for φ_n, and (4) is exactly the statement that φ_n(0) does not decay. The Shapiro–Shields condition is recovered as a special case (Remark 3.4) where all μ_n are replaced by arc measure. Section 4's sufficient conditions and the zero-divisor corollary look solid; the Carleson-formula estimates track correctly, and I didn't find any free parameters or invented entities. The paper is honest about the recursive character of the criterion.

The soft spots, in order. (1) The 0 problem. The theorem says any Z⊂D. But if z_n=0, then k^{μ_{n-1}}_0(0)=1, so the sum in (4) diverges; yet {0} is a zero set. The proof's 'without loss of generality, 0∉Z' is not a WLOG—it excludes exactly the cases where (4) fails. Easy fix: state the theorem for sequences in D\{0} and note that 0 can be added to a zero set freely (multiply by z). But as written, the theorem is false. (2) The 'only if' direction relies on [6, Prop 5.2] without stating its hypotheses. A referee will need that proposition stated, and the proof must make clear which ambient D(μ) it's applied to (presumably μ=m). This may be routine for the authors, but it's load-bearing and currently opaque. (3) Minor: the notation alternates between D and D(μ) in the proof of Theorem 3.1; I'd clean that up.

Bottom line: this deserves a serious referee. The main idea is strong, the Section 4 results are useful, and the flaws are presentation-level rather than conceptual. I'd take it conditional on fixing the 0 gap and stating the cited proposition. If those get fixed, I'd be comfortable declaring a real result.

Referee Report

3 major / 5 minor

Summary. The manuscript studies zero sets of the classical Dirichlet space D. Its main result, Theorem 3.1, asserts that a sequence Z=(z_n)_{n≥1}⊂D is a zero set for D if and only if ∑_{n≥1}1/k^{μ_{n-1}}_{z_n}(z_n)<∞, where dμ_n=|φ_n(ζ)|²dm and φ_n is an extremal function for the invariant subspace M(Z_n,D). The proof factorizes φ_n into one-point extremal factors via Lemma 2.2, reduces the criterion to the non-vanishing of lim φ_n(0), and invokes a weak-compactness argument for sufficiency and [6, Proposition 5.2] for necessity. Sections 4 and 4.1 give sufficient conditions for zero sets and zero divisors based on Carleson's formula.

Significance. If Theorem 3.1 is correct after the issues below are fixed, it would provide the first complete characterization of Dirichlet-space zero sets, a long-standing open problem; the classical Shapiro–Shields condition appears as a special case (Remark 3.4). The product formula for φ_n(0) is explicit and gives a concrete, if computable-in-principle, test. The paper also gives quantitative sufficient conditions and connects the criterion to the harmonic Dirichlet spaces D(μ). The main theorem as stated, however, is false for zero sequences containing 0, and the necessity direction is not self-contained because it relies on an unstated proposition from [6]. The core idea is promising and likely repairable, but the paper needs substantial revision before acceptance.

major comments (3)

[Theorem 3.1, proof, first sentence] Theorem 3.1 is false as stated for sequences containing 0. The proof begins “Without loss of generality, we can assume that 0∉Z”, but this is not a harmless reduction for the theorem as written. If z_n=0 for some n, then k^{μ_{n-1}}_0(0)=1, as stated just before formula (3), so the summand in (4) equals 1 and the series diverges. Yet {0} is a zero set of D, witnessed by f(z)=z. The equivalence therefore fails for every enumeration containing 0. The theorem must be restricted to Z⊂D\{0}, or the statement must explicitly reduce the zero-containing case to Z\{0} and explain how the extremal factor for the point 0, which is not given by (3), is handled. This is load-bearing because the theorem is the paper's central claim.
[Theorem 3.1, proof, necessity direction] The 'only if' direction is not self-contained. It invokes [6, Proposition 5.2] to assert that if M=∩M_n≠{0}, then the extremal functions φ_n converge weakly to φ_M, so that lim φ_n(0)>0. The proposition is not stated, and the proof writes the ambient space as D(μ) without specifying μ. The reader must verify that the proposition applies to the decreasing family M(Z_n,D) inside D=D(m), and that the conclusion about φ_n(0) follows. Since this is the load-bearing step of necessity, please state the proposition, check its hypotheses in the present setting, and supply the short argument that weak convergence of φ_n to φ_M gives φ_n(0)→φ_M(0)>0. If [6, Proposition 5.2] requires additional hypotheses on μ, those must be verified explicitly.
[Theorem 3.1, condition (4)] As written, condition (4) depends on the enumeration of Z through the measures μ_{n-1}, whereas the property of being a zero set is independent of the enumeration. The theorem should state explicitly whether (4) is required for the given enumeration or for every enumeration, and the proof should either justify the independence or explain that the argument works for any enumeration. This is related to the use of [6, Proposition 5.2], but a sentence in the statement would remove a genuine ambiguity for the reader.

minor comments (5)

[Theorem 4.1, after (10)] The displayed estimate appears to read '|φn(ζ)|' where the intended expression is '|φ(ζ)|^N' (or '|φ^N(ζ)|'); please correct the notation to avoid confusion with the extremal functions φ_n.
[Theorem 4.1, proof] The parameter γ is used in three successive roles: first as an auxiliary parameter, then under the condition γ>α, and finally under γ>α+M. Renaming these parameters would make the chain of inequalities much easier to follow.
[Section 2 and Theorem 3.1] The measure dμ_n is defined both in Section 2 and in the statement of Theorem 3.1; unify these definitions and make sure k^{μ_{n-1}}_{z_n} is defined before its first appearance in (4).
[Introduction] The phrase 'zero set' is used without a formal definition. For a sequence Z, please specify whether repetitions are allowed and whether the condition is that a function in D vanishes on Z, at least on Z, or exactly on Z.
[Remark 3.3] The claim that Theorem 3.1 extends to weighted Dirichlet spaces D_α would be more useful if the exact form of condition (4) in that generality were stated, since the measures μ_n depend on the ambient space.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central criterion is a genuinely new recursive condition and the cited convergence theorem is independent support.

full rationale

The paper's Theorem 3.1 gives a recursive Shapiro–Shields-type criterion (4) built from the extremal functions φ_n of the vanishing subspaces M(Z_n, D). The proof first expresses φ_n(0) by formula (3) as the product ∏(1 − 1/k^{μ_{j−1}}_{z_j}(z_j))^{1/2}, so condition (4) is exactly the condition that φ_n(0) does not tend to 0. This equivalence is a direct algebraic manipulation, not a definitional identification of the criterion with the zero-set property. The remaining step—that ∩_n M_n ≠ {0} is equivalent to lim φ_n(0) > 0—is supplied by [6, Proposition 5.2], a published theorem by an overlapping author (El-Fallah). The argument does not reduce to a self-citation chain: [6] is an independent, citable theorem about extremal functions for decreasing families of invariant subspaces, and the present paper's contribution is the explicit kernel criterion (4) built from it. No fitted parameters are renamed as predictions, and no uniqueness claim is imported to forbid alternatives. The only concern detected is a correctness edge case: the proof's 'WLOG 0∉Z' is not a genuine WLOG, since formula (3) is undefined at a = 0 and the stated criterion diverges for sequences containing 0 even though such sequences can be zero sets; this is a mathematical flaw, not circularity, and does not affect the circularity score.

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

The criterion (4) is defined from the extremal functions of the vanishing subspaces of the sequence itself, so the theorem's content is the equivalence between this recursive analytic condition and non-triviality of ∩ M(Z_n, D). The mathematical load is carried by published results on invariant subspaces, chiefly the authors' own [6, Prop 5.2], plus the product formula derived in Section 2 and formula (3). No free parameters are fitted and no entities are invented; the entrance fee is the cited invariant-subspace machinery, which a reader must accept on the strength of the earlier publications.

assumptions (5)

domain assumption Richter's representation theorem for shift-invariant subspaces of D(μ): M = φ_M D(μ_{φ_M}) with dμ_{φ_M} = |φ_M|² dμ and isometric factor map (Richter [14, Theorem 7.1]).
Foundational for Lemma 2.1 and Lemma 2.2, which give the product structure of extremal functions used throughout Section 3 and in condition (4).
domain assumption [6, Prop 5.2]: for a decreasing family of shift-invariant subspaces (M_n) of D(μ) with ∩_n M_n ≠ {0}, the associated extremal functions converge weakly to the extremal function of the intersection, hence lim φ_n(0) = φ_M(0) > 0.
Load-bearing premise for the necessity direction of Theorem 3.1; cited but neither stated nor proved in this paper.
domain assumption Carleson's formula: D(Bf) = D(f) + (1/2π) Σ_{z∈Z} ∫_T ((1-|z|²)/|1-ζ̄z|²)|f(ζ)|²|dζ| for a Blaschke product B with zero set Z (Carleson [3]; see [7, Theorem 4.1.3]).
Key tool in the proofs of Theorem 4.1, Corollary 4.2, and Theorem 4.3.
domain assumption Extremal functions of D(μ) are multipliers with |φ(z)| ≤ 1 on D (Aleman [1, Lemma 4.8]; Shimorin [21, Theorem 1]).
Used implicitly in the normality argument of Theorem 3.1 and explicitly (∥φ∥_∞ ≤ 1) in the estimates of Section 4.
domain assumption Kernel asymptotics k^μ_z(z) ≍ 1 + ∫_0^{|z|} dr/((1-r)Pμ(rz/|z|) + (1-r)²), uniformly in z and μ (El-Fallah, Elmadani, Kellay [5, Theorem 1]).
Remark 3.2 presents this as the way to apply criterion (4); it is the authors' own prior result and is not derived in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Extremal functions and zero sets for the Dirichlet space." pith.science (2026). https://pith.science/paper/7Q4CJEZC

@misc{pith2026260814103,
  author       = {Pith},
  title        = {Pith review of: Extremal functions and zero sets for the Dirichlet space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q4CJEZC}},
  note         = {Machine review of arXiv:2608.14103}
}

read the original abstract

We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[6]

El-Fallah, Y

O. El-Fallah, Y. Elmadani, I. Labghail,Extremal functions and invariant subspaces in Dirichlet spaces, Adv. Math.408(2022), part B, Paper No. 108604, 29 pp

work page 2022
[1]

Aleman,The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, FernUniversit¨ at Hagen, 1993

A. Aleman,The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, FernUniversit¨ at Hagen, 1993

work page 1993
[2]

Arcozzi; R

N. Arcozzi; R. Rochberg; E. Sawyer; B. Wick. The Dirichlet space and related function spaces. Mathematical Surveys and Monographs, 239. American Mathematical Society, Providence, RI, 2019

work page 2019
[3]

Carleson,A representation formula for the Dirichlet integral, Math

L. Carleson,A representation formula for the Dirichlet integral, Math. Z.73(1960) 190–196

work page 1960
[4]

Carleson,On the zeros of functions with bounded Dirichlet integrals, Math

L. Carleson,On the zeros of functions with bounded Dirichlet integrals, Math. Z.56(1952) 289–295

work page 1952
[5]

El-Fallah, Y

O. El-Fallah, Y. Elmadani, K. Kellay,Kernel estimate and capacity in Dirichlet space, J. Funct. Anal. 276(2019) 867–895. 12 A. BORICHEV, O. EL-F ALLAH, AND K. KELLAY

work page 2019
[7]

El-Fallah, K

O. El-Fallah, K. Kellay, J. Mashreghi, T. Ransford,A primer on the Dirichlet space, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2014

work page 2014
[8]

Cheng,Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? J

C. Cheng,Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? J. Funct. Anal.279(2020) Paper No. 108608, 15 pp

work page 2020

Show all 22 references

[9]

Hartz, S

M. Hartz, S. Richter,Inner factors of Dirichlet space functions, ArXiv:2512.17723
[10]

Kellay, J

K. Kellay, J. Mashreghi,On zero sets in the Dirichlet space, J. Geom. Anal.22(2012) 1055–1070

2012
[11]

Nagel, W

A. Nagel, W. Rudin, J. Shapiro,Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. Math.116(1982) 331–360

1982
[12]

Richter,Invariant subspaces in Banach spaces of analytic functions, Trans

S. Richter,Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc.304 (1987) 585–616

1987
[13]

Richter,Invariant subspaces of the Dirichlet shift, J

S. Richter,Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math.386(1988) 205–220

1988
[14]

Richter,A representation theorem for cyclic analytic two-isometries, Trans

S. Richter,A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc.328 (1991) 325–349

1991
[15]

Richter, W

S. Richter, W. T. Ross, C. Sundberg,Zeros of functions with finite Dirichlet integrals, Proc. Amer. Math. Soc.132(2004) 2361–2365

2004
[16]

Richter, C

S. Richter, C. Sundberg,A Formula for the local Dirichlet integral, Mich. Math. J.38(1991) 355–379

1991
[17]

Richter, C

S. Richter, C. Sundberg,Multipliers and invariant subspaces in the Dirichlet space, J. Oper. Th.28 (1992) 167–186

1992
[18]

Richter, C

S. Richter, C. Sundberg,Invariant subspaces of the Dirichlet shift and pseudocontinuations, Trans. Amer. Math. Soc.341(1994) 863–879

1994
[19]

Seip,Interpolating and sampling in spaces of analytic functions, American Mathematical Society, Providence, 2004

K. Seip,Interpolating and sampling in spaces of analytic functions, American Mathematical Society, Providence, 2004

2004
[20]

H. S. Shapiro, A. L. Shields,On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z.80(1962) 217–299

1962
[21]

Shimorin,Reproducing kernels and extremal functions in Dirichlet-type spaces, Zap

S. Shimorin,Reproducing kernels and extremal functions in Dirichlet-type spaces, Zap. Nauchn. Sem. POMI255(1998) 198–220; English translation in: J. Math. Sci.107(2001), no. 4, 4108–4124

1998
[22]

Shimorin,Complete Nevanlinna–Pick property of Dirichlet-type spaces, J

S. Shimorin,Complete Nevanlinna–Pick property of Dirichlet-type spaces, J. Funct. Anal.191(2002) 276–296. A. Borichev, Aix-Marseille University, CNRS, I2M, Marseille, France Email address:alexander.borichev@math.cnrs.fr O. El-F allah, Laboratory of Mathematical Analysis and Ap...

2002

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [6]

El-Fallah, Y

O. El-Fallah, Y. Elmadani, I. Labghail,Extremal functions and invariant subspaces in Dirichlet spaces, Adv. Math.408(2022), part B, Paper No. 108604, 29 pp

work page 2022

[2] [1]

Aleman,The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, FernUniversit¨ at Hagen, 1993

A. Aleman,The multiplication operator on Hilbert spaces of analytic functions, Habilitationsschrift, FernUniversit¨ at Hagen, 1993

work page 1993

[3] [2]

Arcozzi; R

N. Arcozzi; R. Rochberg; E. Sawyer; B. Wick. The Dirichlet space and related function spaces. Mathematical Surveys and Monographs, 239. American Mathematical Society, Providence, RI, 2019

work page 2019

[4] [3]

Carleson,A representation formula for the Dirichlet integral, Math

L. Carleson,A representation formula for the Dirichlet integral, Math. Z.73(1960) 190–196

work page 1960

[5] [4]

Carleson,On the zeros of functions with bounded Dirichlet integrals, Math

L. Carleson,On the zeros of functions with bounded Dirichlet integrals, Math. Z.56(1952) 289–295

work page 1952

[6] [5]

El-Fallah, Y

O. El-Fallah, Y. Elmadani, K. Kellay,Kernel estimate and capacity in Dirichlet space, J. Funct. Anal. 276(2019) 867–895. 12 A. BORICHEV, O. EL-F ALLAH, AND K. KELLAY

work page 2019

[7] [7]

El-Fallah, K

O. El-Fallah, K. Kellay, J. Mashreghi, T. Ransford,A primer on the Dirichlet space, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2014

work page 2014

[8] [8]

Cheng,Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? J

C. Cheng,Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? J. Funct. Anal.279(2020) Paper No. 108608, 15 pp

work page 2020