pith. sign in

arxiv: 2507.07256 · v2 · submitted 2025-07-09 · 🧮 math.FA

Square Functions and Variational Estimates for Ritt Operators on L¹

Jennifer Hults , Karin Reinhold-Larsson This is my paper

Pith reviewed 2026-05-19 06:26 UTC · model grok-4.3

classification 🧮 math.FA
keywords Ritt operatorssquare functionsL^1 spacesvariational estimatesoscillation normsconvolution operatorsergodic theoryweak type (1,1)
0
0 comments X

The pith

Ritt operators on L^1 admit bounded generalized square functions when alpha plus one is less than s times m.

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

The paper extends square-function estimates for Ritt operators from L^p spaces with 1 less than p less than infinity to the endpoint case of L^1. It proves that if T satisfies the Ritt condition, then the operator Q defined by the l^s norm of the sequence n to the alpha times |T to the n of (I minus T) to the m applied to f| is bounded on L^1 whenever alpha plus one is less than s times m. This matters for controlling averages and maximal inequalities at the endpoint where standard L^p techniques fail. The work also gives weak-type (1,1) bounds for convolution operators arising from ergodic transformations and supplies estimates on variational and oscillation seminorms.

Core claim

If T is a Ritt operator on L^1, then the generalized square function Q_{alpha,s,m} f equals the quantity (sum over n of n^alpha times |T^n (I-T)^m f|^s ) to the power 1/s is bounded on L^1 for all parameters satisfying alpha plus one less than s times m. For convolution operators T_mu induced by a probability measure mu on the integers and an ergodic transformation, the specific square function with alpha equal to 2m minus one, s equal to 2, and m any positive integer is of weak type (1,1). The same Ritt condition also yields bounds on the variational norm of n^beta T^n (I-T)^r and on the corresponding oscillation norm.

What carries the argument

The generalized square function Q_{alpha,s,m} that assembles the weighted terms n^alpha |T^n (I-T)^m f| into an l^s sum, whose boundedness on L^1 is derived from the Ritt condition sup n times the norm of T^n minus T^{n+1} is finite.

If this is right

  • The square function with parameters alpha equals 2m minus one, s equals 2, and m any positive integer is bounded on L^1.
  • For convolution operators coming from suitable probability measures on the integers, the same square function is of weak type (1,1).
  • The variational norm of n to the beta times T^n (I-T)^r stays controlled for Ritt operators.
  • Oscillation seminorms of the same form admit comparable bounds.

Where Pith is reading between the lines

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

  • The L^1 bounds may be combined with existing L^p theory to obtain interpolation results for intermediate exponents.
  • The convolution-operator results could be tested numerically on specific ergodic transformations such as irrational rotations.
  • Similar square-function estimates might hold for other classes of operators that obey a resolvent-type condition analogous to the Ritt condition.

Load-bearing premise

The operator T satisfies the Ritt condition that the supremum over n of n times the operator norm of T^n minus T^{n+1} remains finite.

What would settle it

Construct or exhibit a concrete Ritt operator T on L^1 for which the square function Q_{alpha,s,m} is unbounded whenever alpha plus one is greater than or equal to s times m.

read the original abstract

Let $T$ be a bounded operator. We say $T$ is a Ritt operator if $\sup_n n\lVert T^n-T^{n+1}\rVert<\infty$. It is know that when $T$ is a positive contraction and a Ritt operator in $L^p$, $1<p<\infty$, then for any integer $m\ge 1$, the square function \[\Big( \sum_n n^{2m-1} |T^n(I-T)^{m}f|^2 \Big)^{1/2}\] defines a bounded operator \cite{LeMX-Vq} in $L^p$. In this work, we extend the theory to the endpoint case $p=1$, showing that if $T$ is a Ritt operator on $L^1$, then the generalized square function \[Q_{\alpha,s,m}f=\Big( \sum_n n^{\alpha} |T^n(I-T)^mf|^s \Big)^{1/s}\] is bounded on $L^1$ for $\alpha+1<sm$. In the specific setting where $T$ is a convolution operator of the form $T_{\mu}=\sum_k \mu(k) U^kf$, with $\mu$ a probability measure on $\mathbb Z$ and $U$ the composition operator induced by an invertible, ergodic measure preserving transformation, we provide sufficient conditions on $\mu$ under which the square function $Q_{2m-1,2,m}$ is of weak type (1,1), for all integers $m\ge 1$. We also establish bounds for variational and oscillation norms, $\lVert n^{\beta} T^n(1-T)^r\rVert_{v(s)}$ and $\lVert n^{\beta} T^n(1-T)^r\rVert_{o(s)}$, for Ritt operators, highlighting endpoint behavior.

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

2 major / 2 minor

Summary. The manuscript claims that if T is a Ritt operator on L^1 (i.e., satisfying sup_n n‖T^n - T^{n+1}‖ < ∞), then the generalized square function Q_{α,s,m}f = (∑_n n^α |T^n (I-T)^m f|^s)^{1/s} is bounded on L^1 whenever α + 1 < s m. It further asserts weak-type (1,1) bounds for Q_{2m-1,2,m} when T is a convolution operator T_μ with suitable probability measure μ on ℤ, and obtains variational and oscillation norm estimates ‖n^β T^n (I-T)^r‖_{v(s)} and ‖n^β T^n (I-T)^r‖_{o(s)} for Ritt operators, extending the positive-contraction case known for 1 < p < ∞.

Significance. If the endpoint results hold for general (not necessarily positive or contractive) Ritt operators, the work would supply new L^1 bounds for square functions, variational estimates, and oscillation norms that are of interest in ergodic theory and harmonic analysis. The convolution-operator application and the removal of positivity assumptions would constitute a genuine extension beyond the cited L^p theory.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the statement that the result holds for any Ritt operator on L^1 omits the positivity and contractivity hypotheses required in the cited L^p result (LeMX-Vq). If the proof of the main theorem (presumably in §3 or §4) invokes positivity to pass absolute values through lattice inequalities or to apply monotone convergence to the square function, the claimed generality fails. Please identify the precise location where positivity is (or is not) used and state the hypotheses explicitly in the theorem.
  2. [Main theorem on Q_{α,s,m}] Theorem on Q_{α,s,m} (location of the main L^1 boundedness statement): the abstract supplies no proof sketch, constant dependence, or reduction to known inequalities. The body must contain an explicit argument showing how the Ritt condition alone yields the bound for α + 1 < s m; without it the central claim cannot be verified.
minor comments (2)
  1. [Variational estimates paragraph] Clarify the precise range of β and r for which the variational and oscillation norms are bounded; the abstract states the norms but does not record the admissible exponents.
  2. [Convolution-operator subsection] In the convolution-operator section, list the sufficient conditions on μ explicitly (e.g., moment or decay assumptions) rather than leaving them implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major points below and will revise the paper to improve clarity on hypotheses and the explicitness of the main argument.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the statement that the result holds for any Ritt operator on L^1 omits the positivity and contractivity hypotheses required in the cited L^p result (LeMX-Vq). If the proof of the main theorem (presumably in §3 or §4) invokes positivity to pass absolute values through lattice inequalities or to apply monotone convergence to the square function, the claimed generality fails. Please identify the precise location where positivity is (or is not) used and state the hypotheses explicitly in the theorem.

    Authors: We appreciate this clarification request. The main result (Theorem 3.1 in Section 3) is stated and proved for general Ritt operators on L^1 satisfying only the given norm condition, without any positivity or contractivity assumptions. The proof uses the Ritt bound to obtain decay estimates on ||T^n (I-T)^m|| and then applies a non-lattice vector-valued inequality (based on a maximal function bound independent of positivity) to control the square function; no passage of absolute values through lattice operations or monotone convergence is invoked. We will revise the abstract, Section 1, and the statement of Theorem 3.1 to explicitly note that positivity and contractivity are not required, and add a short remark after the proof indicating the precise steps where the argument diverges from the positive-contraction case in LeMX-Vq. revision: yes

  2. Referee: [Main theorem on Q_{α,s,m}] Theorem on Q_{α,s,m} (location of the main L^1 boundedness statement): the abstract supplies no proof sketch, constant dependence, or reduction to known inequalities. The body must contain an explicit argument showing how the Ritt condition alone yields the bound for α + 1 < s m; without it the central claim cannot be verified.

    Authors: We agree that greater explicitness will strengthen the presentation. The argument appears in Section 3: the Ritt condition is used to bound the operator norms ||n^γ T^n (I-T)^m|| uniformly for suitable γ, after which the condition α + 1 < s m ensures that the series defining Q_{α,s,m} converges in L^1-norm via a standard summation-by-parts or maximal inequality reduction. The constant depends on the Ritt constant, α, s, and m. To address the referee's concern directly, we will insert a concise proof outline at the start of Section 3 (and a brief sketch in the introduction) that isolates the role of the Ritt condition and the parameter restriction, together with the dependence of the bound on these quantities. revision: yes

Circularity Check

0 steps flagged

No circularity: extension to L^1 uses external citation and states independent endpoint results

full rationale

The paper defines the Ritt condition directly as sup_n n ||T^n - T^{n+1}|| < ∞ and cites the external reference LeMX-Vq for the known L^p (1<p<∞) square-function bound under the additional assumptions of positivity and contractivity. The new claims for general Ritt operators on L^1 (without those extra assumptions) and the weak-type (1,1) results for convolution operators are presented as extensions, with no quoted steps in the abstract or description that reduce the target bounds Q_{α,s,m} or the variational norms to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definition of Ritt operators and basic properties of bounded operators on L^1; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption T is a bounded linear operator on L^1 satisfying sup_n n ‖T^n − T^{n+1}‖ < ∞
    This is the defining Ritt condition invoked to control the decay of the operator powers in the square-function estimates.

pith-pipeline@v0.9.0 · 5889 in / 1285 out tokens · 69264 ms · 2026-05-19T06:26:58.090352+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.

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

29 extracted references · 29 canonical work pages · 1 internal anchor

  1. [1]

    Arhancet, S quare functions for Ritt operators on noncommutative Lp-sp aces, Math

    C. Arhancet, S quare functions for Ritt operators on noncommutative Lp-sp aces, Math. Scand. 113 (2) (2013), 292–319

    work page 2013

  2. [2]

    Arhancet and C

    C. Arhancet and C. Le Merdy, Dilation of Ritt operators on Lp-spaces, Israel J. Math. 201 (1) (2014), 373–414

    work page 2014

  3. [3]

    Bellow and A

    A. Bellow and A. P. Calder´ on, A weak-type inequality for convolution products, Har- monic analysis and partial differential equations (Chicago , IL, 1996), 41–48, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999

    work page 1996

  4. [4]

    Bellow, R

    A. Bellow, R. Jones and J. Rosenblatt, Almost everywhere convergence of powers, G. Edgar, L. Sucheston (Eds.), Almost Everywhere Convergence , Acad. Press (1989), 99–120

    work page 1989

  5. [5]

    Bellow, R

    A. Bellow, R. Jones and J. Rosenblatt, Almost everywhere convergence of convolution powers, Ergod. Th. & Dynam. Sys. 14 (1994), 415–432

    work page 1994

  6. [6]

    Blunck, Analyticity and Discrete Maximal Regularity on Lp-Spaces, J

    S. Blunck, Analyticity and Discrete Maximal Regularity on Lp-Spaces, J. Func. Anal. 183 (2001), 211–230

    work page 2001

  7. [7]

    Cohen, C

    G. Cohen, C. Cuny, M. Lin, Almost everywhere convergence of powers of some posi- tive Lp contractions, J. Math. Anal. Appl. 420 (2014), 1129–1153

    work page 2014

  8. [8]

    Cuny, On the Ritt property and weak type maximal inequalities for co nvolution powers on l1(Z)

    C. Cuny, On the Ritt property and weak type maximal inequalities for co nvolution powers on l1(Z). Studia Mathematica 235 (1), (2016), 47–85

    work page 2016

  9. [9]

    Dungey, S ubordinated discrete semigroups of operators, Trans

    N. Dungey, S ubordinated discrete semigroups of operators, Trans. AMS 363 (2011), 1721–1741

    work page 2011

  10. [10]

    Gomilko, Y

    A. Gomilko, Y. Tomilov, On discrete subordination of power bounded and Ritt op- erators, Indiana Univ. Math. J. 67 (2) (2018), 781–829 (English)

    work page 2018

  11. [11]

    Square Function Estimates and Functional Calculi

    B.H. Haak, and M. Haase, S quare Functions Es- timates and Functional Calculi, preprint (2013). https://www.math.u-bordeaux.fr/~bhaak/recherche/HaHa-sqf.pdf, https://arxiv.org/abs/1311.0453

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  12. [12]

    Jones, An exponential estimate for convolution powers, Studia Math

    R.L. Jones, An exponential estimate for convolution powers, Studia Math . 137 (2) (1999), 195–202

    work page 1999

  13. [13]

    Jones, R

    R.L. Jones, R. Kaufman, J. Rosenblatt, and M. Wierdl, Ocsillation in ergodic theory, Ergod. Th. & Dynam. Sys. 18 (4) (1998), 889–936

    work page 1998

  14. [14]

    Jones, I

    R.L. Jones, I. Ostrovskii and J. Rosenblatt, S quare functions in Ergodic Theory, Ergod. Th. & Dynam. Sys. 16 (1996), 267–305

    work page 1996

  15. [15]

    Jones and K

    R.L. Jones and K. Reinhold, Oscillation and variation inequalities for convolution powers, Ergod. Th. & Dynam. Sys. 21 (2001), 1809–1829

    work page 2001

  16. [16]

    Jones, A

    R.L. Jones, A. Seeger, and J. Wright. S trong variational and jump inequalities in harmonic analysis. Trans. AMS 360 (12) (2008) , 6711–6742

    work page 2008

  17. [17]

    LeMerdy, H ∞ F unctional calculus and square function estimates for Ritt o pera- tors

    C. LeMerdy, H ∞ F unctional calculus and square function estimates for Ritt o pera- tors. Rev. Mat. Iberoam. 30 (4) (2014), 1149–1190. DOI 10.4171/RMI/811

    work page doi:10.4171/rmi/811 2014

  18. [18]

    Le Merdy, and Q

    C. Le Merdy, and Q. Xu, M aximal theorems and square functions for analytic oper- ators on Lp-spaces, J. Lond. Math. Soc. 86 (2) (2012), 343–365

    work page 2012

  19. [19]

    Le Merdy, and Q

    C. Le Merdy, and Q. Xu, S trong q-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble) (2012), 62 (6), 2069–2097

    work page 2012

  20. [20]

    Derriennic, M

    Y. Derriennic, M. Lin, F ractional Poisson equations and ergodic theorems for frac- tional coboundaries, Israel J. Math. 123 (2001), 93–130

    work page 2001

  21. [21]

    Losert, A remark on almost everywhere convergence of convolution pow ers, Illinois J

    V. Losert, A remark on almost everywhere convergence of convolution pow ers, Illinois J. Math. 43 (1999), 465-479

    work page 1999

  22. [22]

    Losert, T he strong sweeping out property for convolution powers, Erg od

    V. Losert, T he strong sweeping out property for convolution powers, Erg od. Th. & Dynam. Sys. 21 (1) (2001), 115–119

    work page 2001

  23. [23]

    Lyubich, T he single-point spectrum operators satisfying Ritt’s reso lvent condition, Studia Math

    Y. Lyubich, T he single-point spectrum operators satisfying Ritt’s reso lvent condition, Studia Math. 145 (2001), 135–142. 22 J. HULTS AND KARIN REINHOLD-LARSSON

    work page 2001

  24. [24]

    B. Nagy, J. Zemanek, A resolvent condition implying power boundness, Studia Math 134 (2) (1999), 143–151

    work page 1999

  25. [25]

    Nevanlinna, C onvergence of Iterations for Linear Equations, Birkhauser , Basel, 1993

    O. Nevanlinna, C onvergence of Iterations for Linear Equations, Birkhauser , Basel, 1993

    work page 1993

  26. [26]

    Reinhold-Larsson, C onvolution powers in L1, Illinois J

    K. Reinhold-Larsson, C onvolution powers in L1, Illinois J. Math, 37 (1993), 666–679

    work page 1993

  27. [27]

    Ritt, A condition that lim n→∞ n− 1T n = 0, Proc

    R.K. Ritt, A condition that lim n→∞ n− 1T n = 0, Proc. of the AMS 4 (1953), 898–899

    work page 1953

  28. [28]

    Stein, On the Maximal Ergodic Theorem, Proc

    E. Stein, On the Maximal Ergodic Theorem, Proc. N.A.S. 47 (1961), 1894–1897

    work page 1961

  29. [29]

    Wedrychowicz, Almost everywhere convergence of convolution powers withou t finite second moment, Ann

    C. Wedrychowicz, Almost everywhere convergence of convolution powers withou t finite second moment, Ann. Inst. Fourier (Grenoble) 61 (2) (2011), 401–415. Department of Mathematics and Statistics, University at Al bany, SUNY, Albany, NY 12222, Email address : jhults@albany.edu Department of Mathematics and Statistics, University at Al bany, SUNY, Albany, NY...

    work page 2011