Constructive description of analytic Besov spaces in strictly pseudoconvex domains

Aleksandr Rotkevich

arxiv: 1907.01584 · v1 · pith:NCFWYHFEnew · submitted 2019-07-02 · 🧮 math.CV

Constructive description of analytic Besov spaces in strictly pseudoconvex domains

Aleksandr Rotkevich This is my paper

Pith reviewed 2026-05-25 10:15 UTC · model grok-4.3

classification 🧮 math.CV

keywords analytic Besov spacesstrictly pseudoconvex domainspseudoanalytic continuationpolynomial approximationsholomorphic functionsboundary valuesseveral complex variables

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The pith

Holomorphic functions in strictly pseudoconvex domains with Besov boundary values are characterized by their rates of polynomial approximation.

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

The paper applies the method of pseudoanalytic continuation to characterize spaces of holomorphic functions whose boundary values lie in Besov spaces. The characterization is expressed through the rates at which these functions can be approximated by polynomials. This yields a constructive description of the analytic Besov spaces on the boundary. A reader would see this as connecting interior approximation properties directly to boundary regularity in several complex variables.

Core claim

The method of pseudoanalytic continuation produces a characterization of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations, valid in strictly pseudoconvex domains.

What carries the argument

The method of pseudoanalytic continuation, which extends the holomorphic functions so that their polynomial approximation rates encode the Besov norm of the boundary values.

If this is right

The Besov norm on the boundary is equivalent to a norm defined by the rate of polynomial approximation inside the domain.
The characterization holds for all strictly pseudoconvex domains.
Polynomial approximations provide an explicit, constructive way to measure membership in these analytic Besov spaces.

Where Pith is reading between the lines

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

The same continuation technique could be tested on other function spaces whose norms involve derivatives or fractional smoothness.
The approximation characterization might simplify proofs that certain operators are bounded on these spaces.

Load-bearing premise

The method of pseudoanalytic continuation applies directly to holomorphic functions in strictly pseudoconvex domains and yields the stated polynomial-approximation characterization of the corresponding Besov boundary spaces.

What would settle it

For a concrete holomorphic function in the unit ball whose boundary values have a known Besov norm, compute the polynomial approximation errors and check whether they satisfy the exact rate predicted by the Besov index.

read the original abstract

We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.

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

Rotkevich claims a characterization of analytic Besov spaces in strictly pseudoconvex domains via pseudoanalytic continuation and polynomial approximations, but the abstract supplies almost no supporting steps.

read the letter

Hi, the main takeaway is that this paper uses pseudoanalytic continuation to link holomorphic functions whose boundary values sit in Besov spaces to rates of polynomial approximation, all inside strictly pseudoconvex domains. That is the stated result. What looks new is the concrete combination of the continuation technique with polynomial approximation to produce an explicit description of the analytic Besov spaces in this geometry. The abstract frames it as a constructive characterization, which is the angle the author is pushing. The work does a reasonable job of identifying a narrow technical target where such a link could be useful for people already tracking boundary behavior of holomorphic functions. It stays within the subfield and does not overclaim broader impact. The clearest soft spot is the complete absence of any estimate, construction detail, or verification step in the abstract. Soundness cannot be checked from what is given, and the low reader is understandable for that reason. It is also impossible to tell from the supplied text whether the argument adds something independent or simply repackages known facts about continuation and approximation. The citation pattern and comparison to prior literature are not visible here either. This is a paper for a small group of specialists who already work on Besov spaces or pseudoanalytic methods in several complex variables. Someone outside that circle will not get much from it. A reader inside the area might find the characterization handy if the details check out. It deserves a serious referee because the claim is well-defined and the setting is standard enough that external eyes could confirm whether the continuation really yields the stated approximation control. My recommendation is to send it to review once the full manuscript is in hand, with the expectation that the proofs and literature placement will need close scrutiny.

Referee Report

0 major / 0 minor

Summary. The paper claims to use the method of pseudoanalytic continuation to characterize spaces of holomorphic functions whose boundary values lie in Besov spaces, in strictly pseudoconvex domains, via polynomial approximations.

Significance. If the claimed characterization holds, it would supply a constructive, approximation-theoretic description of analytic Besov spaces on the boundary of strictly pseudoconvex domains, which could be of interest for questions in several complex variables concerning boundary regularity and approximation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments appear in the report, so there are no individual points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract describes a characterization of holomorphic functions with Besov boundary values via pseudoanalytic continuation and polynomial approximations. No equations, definitions, or citations are supplied in the visible text that reduce a claimed result to its own inputs by construction, rename a fitted quantity as a prediction, or rely on load-bearing self-citations. The method is presented as an external tool applied to the domain, with no evidence of self-definitional loops or ansatz smuggling. This is the expected honest non-finding for a characterization paper whose internal steps cannot be shown to collapse.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are visible in the abstract; the work relies on the pre-existing notions of Besov spaces, strictly pseudoconvex domains, and the method of pseudoanalytic continuation, all treated as standard background.

pith-pipeline@v0.9.0 · 5529 in / 1149 out tokens · 46109 ms · 2026-05-25T10:15:12.012595+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The main result … characterization … in terms of polynomial approximations … {2^m s E_m(f)_p} ∈ l^q

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

L. A. Aizenberg, A. P. Yuzhakov, Integral representations and residues in complex Analysis [in Russian], Moscow (1979)

work page 1979
[2]

V. K. Dzyadyk, Introduction to the theory of uniform approximation of func tions by polynomials [in Russian], Moscow (1977)

work page 1977
[3]

E. M. Dyn’kin, Constructive characterization of S. L. Sobolev and O. V. Bes ov classes, Trudy Mat. Inst. AN SSSR, 155, 41-76 (1981). Analytic Besov spaces 17

work page 1981
[4]

Feﬀerman, E.M

C. Feﬀerman, E.M. Stein, H p spaces of several variables , Acta mathematica, Vol. 129, No. 1, 137-193 (1972)

work page 1972
[5]

H¨ ormander, An Introduction to Complex Analysis in Sev eral Variables, Van Nostrand, Princeton (1966)

L. H¨ ormander, An Introduction to Complex Analysis in Sev eral Variables, Van Nostrand, Princeton (1966)

work page 1966
[6]

G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge, 1934

work page 1934
[7]

Lanzani, E

L. Lanzani, E. M. Stein, Cauchy-type integrals in several complex variables , Bull. Math. Sci., Vol. 3, No. 2, 241-285 (2013)

work page 2013
[8]

Leray, Le calcul diﬀ´ erentiel et int´ egral sur une vari´ at´ a analytique complexe

J. Leray, Le calcul diﬀ´ erentiel et int´ egral sur une vari´ at´ a analytique complexe. (Probl` eme de Cauchy. III.) Bull. Soc. Math. Fr. 87, 81-180 (1959)

work page 1959
[9]

R. M. Range, Holomorphic functions and integral representations in sev eral com- plex variables , Springer Verlag (1986)

work page 1986
[10]

A. S. Rotkevich, Constructive description of the Besov classes in convex dom ains in Cn, Zap. Nauch. Sem. POMI 401, 136-174 (2013)

work page 2013
[11]

Rotkevich A.S., Constructive description of Hardy-Sobolev spaces on stron gly con- vex domains in Cn, J. Math. Anal. Appl, 465, 2, 1025-1038 (2018)

work page 2018
[12]

Rotkevich A.S., External area integral inequality for the Cauchy-Leray-Fa ntappi` e integral, Complex Anal. Oper. Th., 2018

work page 2018
[13]

N. A. Shirokov, Jackson-Bernstein theorem in strictly pseudoconvex domai ns in Cn, Constr. Approx., Vol. 5, No. 1, 455-461 (1989)

work page 1989
[14]

N. A. Shirokov, A direct theorem for strictly convex domains in Cn, Zap. Nauch. Sem. POMI 206, 152-175 (1993)

work page 1993
[15]

E. L. Stout, H p-functions on strictly pseudoconvex domains , Amer. J. Math., Vol. 98, No. 3, 821-852 (1976)

work page 1976

[1] [1]

L. A. Aizenberg, A. P. Yuzhakov, Integral representations and residues in complex Analysis [in Russian], Moscow (1979)

work page 1979

[2] [2]

V. K. Dzyadyk, Introduction to the theory of uniform approximation of func tions by polynomials [in Russian], Moscow (1977)

work page 1977

[3] [3]

E. M. Dyn’kin, Constructive characterization of S. L. Sobolev and O. V. Bes ov classes, Trudy Mat. Inst. AN SSSR, 155, 41-76 (1981). Analytic Besov spaces 17

work page 1981

[4] [4]

Feﬀerman, E.M

C. Feﬀerman, E.M. Stein, H p spaces of several variables , Acta mathematica, Vol. 129, No. 1, 137-193 (1972)

work page 1972

[5] [5]

H¨ ormander, An Introduction to Complex Analysis in Sev eral Variables, Van Nostrand, Princeton (1966)

L. H¨ ormander, An Introduction to Complex Analysis in Sev eral Variables, Van Nostrand, Princeton (1966)

work page 1966

[6] [6]

G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge, 1934

work page 1934

[7] [7]

Lanzani, E

L. Lanzani, E. M. Stein, Cauchy-type integrals in several complex variables , Bull. Math. Sci., Vol. 3, No. 2, 241-285 (2013)

work page 2013

[8] [8]

Leray, Le calcul diﬀ´ erentiel et int´ egral sur une vari´ at´ a analytique complexe

J. Leray, Le calcul diﬀ´ erentiel et int´ egral sur une vari´ at´ a analytique complexe. (Probl` eme de Cauchy. III.) Bull. Soc. Math. Fr. 87, 81-180 (1959)

work page 1959

[9] [9]

R. M. Range, Holomorphic functions and integral representations in sev eral com- plex variables , Springer Verlag (1986)

work page 1986

[10] [10]

A. S. Rotkevich, Constructive description of the Besov classes in convex dom ains in Cn, Zap. Nauch. Sem. POMI 401, 136-174 (2013)

work page 2013

[11] [11]

Rotkevich A.S., Constructive description of Hardy-Sobolev spaces on stron gly con- vex domains in Cn, J. Math. Anal. Appl, 465, 2, 1025-1038 (2018)

work page 2018

[12] [12]

Rotkevich A.S., External area integral inequality for the Cauchy-Leray-Fa ntappi` e integral, Complex Anal. Oper. Th., 2018

work page 2018

[13] [13]

N. A. Shirokov, Jackson-Bernstein theorem in strictly pseudoconvex domai ns in Cn, Constr. Approx., Vol. 5, No. 1, 455-461 (1989)

work page 1989

[14] [14]

N. A. Shirokov, A direct theorem for strictly convex domains in Cn, Zap. Nauch. Sem. POMI 206, 152-175 (1993)

work page 1993

[15] [15]

E. L. Stout, H p-functions on strictly pseudoconvex domains , Amer. J. Math., Vol. 98, No. 3, 821-852 (1976)

work page 1976