pith. sign in

arxiv: 2109.11465 · v3 · submitted 2021-09-23 · 🧮 math.FA · math.OC

Laplace-Carleson embeddings and infinity-norm admissibility

Birgit Jacob , Jonathan R. Partington , Sandra Pott , Eskil Rydhe , Felix L. Schwenninger This is my paper

Pith reviewed 2026-05-24 13:49 UTC · model grok-4.3

classification 🧮 math.FA math.OC
keywords Laplace-Carleson embeddingsCarleson intensityBerezin transformadmissibilityOrlicz spacessemigroup systemscontrol operatorsL^∞
0
0 comments X

The pith

Boundedness of Laplace-Carleson embeddings on L^∞ is characterized by the measure's Carleson intensity and weighted Berezin transform.

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

This paper gives a complete description of when Laplace-Carleson embeddings map L^∞ into the appropriate analytic space in a bounded way. The description uses two quantities attached to the measure: its Carleson intensity and the supremum of a weighted Berezin transform. These embedding results are needed to decide whether a control operator is admissible for a linear diagonal semigroup system when the input functions are essentially bounded. Partial or complete results of the same type are obtained for several Orlicz spaces. The work therefore supplies a tool for infinite-dimensional systems theory that goes beyond the usual L^2 setting.

Core claim

The Laplace-Carleson embedding is bounded on L^∞ precisely when the Carleson intensity of the measure is finite and a suitable weighted Berezin transform of the measure is bounded. Analogous boundedness statements, sometimes with full characterizations, hold on a large class of Orlicz spaces. These facts translate directly into conditions for the infinity-norm admissibility of control operators for linear diagonal semigroup systems.

What carries the argument

Laplace-Carleson embedding of a measure, whose boundedness is decided by Carleson intensity together with a weighted Berezin transform.

If this is right

  • Admissibility of control operators for linear diagonal semigroup systems is characterized for essentially bounded inputs.
  • Boundedness results hold for a large class of Orlicz spaces.
  • Full characterizations are available in some cases for those Orlicz spaces.
  • The focus on L^∞ inputs extends classical admissibility theory beyond square-integrable functions.

Where Pith is reading between the lines

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

  • The separation into intensity and transform conditions may allow independent verification of each part in concrete examples.
  • The embedding criteria could be used to compare admissibility across different input spaces.
  • Related measure conditions might appear in embedding problems outside control theory.

Load-bearing premise

The systems under consideration are linear diagonal semigroup systems.

What would settle it

Constructing a measure for which the Carleson intensity is finite but the weighted Berezin transform is unbounded (or the converse) would show that the stated characterization is incomplete.

Figures

Figures reproduced from arXiv: 2109.11465 by Birgit Jacob, Eskil Rydhe, Felix L. Schwenninger, Jonathan R. Partington, Sandra Pott.

Figure 1
Figure 1. Figure 1: Relation between condition (4) and the boundednes [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

A full characterization of the boundedness of Laplace--Carleson embeddings on $L^\infty$ is provided, in terms of the Carleson intensity of the respective measure and of a suitable weighted Berezin transform of the measure. Moreover, boundedness results, and in some cases full characterizations of boundedness, are proved for a large class of Orlicz spaces. These findings are crucial for characterizing admissibility of control operators for linear diagonal semigroup systems in a variety of contexts. A particular focus is laid on essentially bounded inputs.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a full characterization of the boundedness of Laplace-Carleson embeddings on L^∞ in terms of the Carleson intensity of the measure and a suitable weighted Berezin transform of the measure. It also establishes boundedness results (and in some cases full characterizations) for a large class of Orlicz spaces. These embedding results are applied to characterize admissibility of control operators for linear diagonal semigroup systems, with particular attention to essentially bounded inputs.

Significance. If the stated characterization holds, the work supplies a precise, usable criterion for boundedness of these embeddings that directly supports admissibility questions in infinite-dimensional control theory. The extension to Orlicz spaces broadens applicability beyond L^∞ and L^p settings. The explicit link to diagonal semigroup systems is presented as an application rather than part of the core embedding theorem.

minor comments (2)
  1. [Abstract] The abstract refers to 'a suitable weighted Berezin transform' without a preliminary definition or reference; the introduction or §2 should state the precise definition and its relation to the standard Berezin transform.
  2. Notation for the Carleson intensity and the measure class should be introduced consistently in the first section where the main theorem is stated, to avoid ambiguity when the results are applied to admissibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of our results, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract presents a characterization of boundedness for Laplace-Carleson embeddings on L^∞ via Carleson intensity and a weighted Berezin transform, plus extensions to Orlicz spaces, as an application to diagonal semigroup admissibility. No equations, derivations, or self-citations are visible that would allow reduction of any claimed result to its inputs by construction. The central claim is a stated theorem rather than a fitted or renamed input, and the paper is treated as self-contained against external benchmarks with no load-bearing self-referential steps identifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no information on free parameters, axioms, or invented entities is supplied.

pith-pipeline@v0.9.0 · 5622 in / 1076 out tokens · 22446 ms · 2026-05-24T13:49:30.883996+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    A full characterization of the boundedness of Laplace–Carleson embeddings on L^∞ is provided, in terms of the Carleson intensity of the respective measure and of a suitable weighted Berezin transform of the measure.

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.

Forward citations

Cited by 1 Pith paper

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

  1. Implications of structured continuous maximal regularity

    math.FA 2026-05 unverdicted novelty 7.0

    Structured continuous maximal regularity sharpens via weak compactness of convolution operators, delivering new proofs for L1-maximal regularity, Baillon's theorem extensions, and resolution of an abstract input-to-st...

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bennett and R

    C. Bennett and R. Sharpley, Interpolation of operators. Academic Press, Inc., Boston, MA, 1988

    work page 1988

  2. [2]

    Garnett, Bounded Analytic Functions

    J.B. Garnett, Bounded Analytic Functions . Graduate Texts in Mathe- matics 236, Springer 2007

    work page 2007

  3. [3]

    Haak, Kontrolltheorie in Banachr¨ aumen und quadratische Ab- sch¨ atzungen

    B. Haak, Kontrolltheorie in Banachr¨ aumen und quadratische Ab- sch¨ atzungen. PhD thesis, Universit¨ at Karlsruhe, 2005

    work page 2005

  4. [4]

    Haak and C

    B. Haak and C. Le Merdy, α -admissibility of observation and control operators. Houston J. Math. 31 (2005), no. 4, 1153–1167. 27

    work page 2005

  5. [5]

    Hansen and G

    S. Hansen and G. Weiss. The operator Carleson measure cri terion for admissibility of control operators for diagonal semigroup s on l2. Systems Control Lett., 16 (1991), no. 3, 219–227

    work page 1991

  6. [6]

    Jacob and J.R

    B. Jacob and J.R. Partington, Admissibility of control a nd observa- tion operators for semigroups: a survey. Current trends in operator theory and its applications , 199–221, Oper. Theory Adv. Appl., 149, Birkh¨ auser, Basel, 2004

    work page 2004

  7. [7]

    Jacob, R

    B. Jacob, R. Nabiullin, J.R. Partington, and F. Schwenni nger, Infinite- dimensional input-to-state stability and Orlicz spaces. SIAM J. Control Optim., 56 (2018), no. 2, 868–889

    work page 2018

  8. [8]

    Jacob, J.R

    B. Jacob, J.R. Partington and S. Pott, On Laplace–Carles on embedding theorems, J. Functional Analysis 264 (2013), 738–814

    work page 2013

  9. [9]

    Jacob, J.R

    B. Jacob, J.R. Partington and S. Pott, Applications of La place– Carleson embeddings to admissibility and controllability. SIAM J. Con- trol. Optim. 52 (2014), 1299–1313

    work page 2014

  10. [10]

    Jacob, F

    B. Jacob, F. L. Schwenninger, and H. Zwart, On continuit y of solutions for parabolic control systems and input-to-state stabilit y. J. Differential Equations, 266 (2019), no. 10, 6284–6306

    work page 2019

  11. [11]

    Mironchenko, C

    A. Mironchenko, C. Prieur, Input-to-State Stability o f Infinite- Dimensional Systems: Recent Results and Open Questions. SIAM Re- view, 62 (2020), no. 3, 19M1291248

    work page 2020

  12. [12]

    Partington, S

    J.R. Partington, S. Pott and R. Zawiski, Laplace-Carle son embeddings on model spaces and boundedness of truncated Hankel and Toep litz operators. Integral Equations Operator Theory 92 (2020), no. 4, Paper No. 37, 15 pp

    work page 2020

  13. [13]

    Rydhe, On Laplace–Carleson embeddings, and L p-mapping proper- ties of the Fourier transform

    E. Rydhe, On Laplace–Carleson embeddings, and L p-mapping proper- ties of the Fourier transform. Ark. Mat 58 (2020), no. 2, 437–457

    work page 2020

  14. [14]

    Salamon, Infinite-dimensional linear systems with u nbounded con- trol and observation: a functional analytic approach

    D. Salamon, Infinite-dimensional linear systems with u nbounded con- trol and observation: a functional analytic approach. Trans. Amer. Math. Soc. 300 (1987), no. 2, 383–431

    work page 1987

  15. [15]

    Tucsnak and G

    M. Tucsnak and G. Weiss, Observation and control for operator semi- groups. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. Birkh¨ auser Verlag, Basel, 2009

    work page 2009

  16. [16]

    G. Weiss. Admissible observation operators for linear semigroups, Israel J. Math. , 65 (1989), 17–43

    work page 1989

  17. [17]

    G. Weiss. Admissibility of unbounded control operator s. SIAM J. Control Optim. , 27 (1989), no. 3, 527–545. 28

    work page 1989

  18. [18]

    G. Weiss. Two conjectures on the admissibility of contr ol operators. In Estimation and control of distributed parameter systems (V orau, 1990), volume 100 of Internat. Ser. Numer. Math. , pages 367–378. Birkh¨ auser, Basel, 1991

    work page 1990

  19. [19]

    Zwart, Sufficient conditions for admissibility

    H. Zwart, Sufficient conditions for admissibility. Systems Control Lett. , 54 (2005), 973–979. 29

    work page 2005