Pith. sign in

REVIEW 1 major objections 5 minor 29 references

Ces\`aro-type operators on mixed norm spaces

T0 review · 1 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper proves that the generalized Cesàro operator $C_{\mu,\beta}$ is bounded from $H(p,q,\gamma_1)$ to $H(p,q,\gamma_2)$ exactly when the defining measure is an $s$-Carleson measure with $s=\beta+\gamma_1-\gamma_2$.

desk verdict A solid unification of Cesàro-type boundedness on mixed norm spaces; the only real issue is the unstated finiteness of the defining measure, which should be fixed before publication. read the letter →

arxiv 2507.20586 v1 pith:CUP7TEK4 submitted 2025-07-28 math.CV

Óscar Blasco , Alejandro Mas This is my paper · ORCID
classification math.CV MSC 47B3830H20
keywords Cesàro-typeoperatorsmixednormspacesCarlesonmeasuresHadamardproductfractionalderivativesweightedBergmanmomentsboundednessof
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 establishes a complete characterization of when a generalized Cesàro operator is bounded on mixed norm spaces of analytic functions. The operator averages an analytic function against a positive measure on the unit interval, and the paper shows that boundedness between $H(p,q,\gamma_1)$ and $H(p,q,\gamma_2)$ holds if and only if the measure's tail mass decays like a power $s=\beta+\gamma_1-\gamma_2$. This single Carleson condition is both necessary and sufficient, and it is independent of $p$ and $q$: boundedness for any one pair forces boundedness for all allowed pairs. The result unifies and extends earlier scattered characterizations for Hardy spaces, weighted Bergman spaces, and Korenblum spaces.

What carries the argument

The key object is the fundamental function $F_\mu(z)=\int_0^1 d\mu(t)/(1-tz)=\sum_{n=0}^\infty \mu_n z^n$, whose coefficients are the moments of the measure. The operator factors in two ways, $C_{\mu,\beta}f=F_\mu*(fK_{\beta-1})=D^\beta F_\mu*C_{\beta-1}f$, where $*$ is the Hadamard product, $K_{\beta-1}(z)=(1-z)^{-\beta}$, and $D^\beta$ is a fractional derivative. The $s$-Carleson condition $\mu([r,1))\lesssim(1-r)^s$, equivalently $\mu_n=O((n+1)^{-s})$, is translated by Theorem 5.2 into the statement that a fractional derivative $D^\alpha F_\mu$ lies in a mixed norm space $H(p,\infty,\gamma)$; mixed-norm inequalities for Hadamard products then transfer that membership into boundedness of the whole operator.

What would settle it

Take a finite positive measure with $\mu([r,1))\asymp(1-r)^s$ for an exponent $s$ strictly smaller than $\beta+\gamma_1-\gamma_2$, for instance $d\mu(t)=(1-t)^{s-1}dt$, and apply $C_{\mu,\beta}$ to the test functions $f_r(z)=(1-rz)^{-1/p-1/q-\gamma_1}$ used in Lemma 6.8; the theorem predicts the norm in $H(p,q,\gamma_2)$ blows up as $r\to 1$, so a bounded result would refute the characterization.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 6.10: for $\gamma_2<\gamma_1+\beta$, $1\le p<\infty$, and $0<q<\infty$, the operator $C_{\mu,\beta}$ maps $H(p,q,\gamma_1)$ into $H(p,q,\gamma_2)$ if and only if $\mu$ is an $s$-Carleson measure with $s=\beta+\gamma_1-\gamma_2$. The same theorem covers $p=\infty$ through the spaces $A^\infty_\gamma$, and it shows that boundedness for one admissible triple $(p,q)$ is equivalent to boundedness for all of them. As corollaries, the paper recovers and generalizes the recent characterization for weighted Bergman spaces: $C_{\mu,\beta}$ maps $A^p_{\alpha_1}$ into $A^q_{\alpha_2}$ for $1\le p\le q<\infty$ exactly when $\mu$ is $s$-Carleson with $s=\beta+(\alpha_1+2)/p-(\alpha_2+2)/q$. When the target space has smaller $q$ than the source, the Carleson condition is replaced by a moment-sequence condition: $C_{\mu,\beta}$ maps $H(p,\infty,\gamma)$ into $H(p,q,\gamma)$ if and only if $D^{\beta+\gamma-1/p'}F_\mu\in H(p,q,\gamma)$.

Load-bearing premise

The whole setup presumes the measure has finite total mass on the unit interval; if it does not, the moments and the integral defining the operator can diverge, and every boundedness statement loses its meaning.

Editorial extensions

If this is right

  • For fixed $p,q$, the operator $C_{\mu,\beta}$ maps $H(p,q,\gamma_1)$ into $H(p,q,\gamma_2)$ exactly when $\mu([r,1))\lesssim(1-r)^{\beta+\gamma_1-\gamma_2}$; no finer information about the measure is needed.
  • Boundedness for a single pair $1\le p<\infty$, $0<q<\infty$ implies boundedness for every admissible pair, because the Carleson exponent does not depend on $p$ or $q$.
  • On weighted Bergman spaces, $C_{\mu,\beta}:A^p_{\alpha_1}\to A^q_{\alpha_2}$ for $1\le p\le q<\infty$ is characterized by $s=\beta+(\alpha_1+2)/p-(\alpha_2+2)/q$, recovering the previously known weighted-Bergman criterion as a special case.
  • If $\mu$ is $s$-Carleson and $\beta>s$, the pointwise estimate $|C_{\mu,\beta}f(z)|\lesssim P^*(f)(z)(1-|z|)^{s-\beta}$ shows the operator shifts the radial weight by $\beta-s$ without changing $p$ or $q$.
  • For $q_1>q_2$ the Carleson criterion fails; instead $C_{\mu,\beta}$ maps $H(p,\infty,\gamma)$ into $H(p,q,\gamma)$ iff $D^{\beta+\gamma-1/p'}F_\mu\in H(p,q,\gamma)$, a moment/sequence-space condition rather than a geometric condition on the measure.

Reading between the lines

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

  • The theorem is phrased through moments and Carleson estimates, so the same characterization is likely to survive for complex Borel measures of finite total variation once the integral representation is interpreted in the sense of the paper's earlier extensions; the proof structure suggests the Carleson condition remains necessary and sufficient.
  • Section 7 shows that varying $q$ interpolates between Carleson-type conditions and Kellogg sequence-space conditions, so an interpolation argument might yield a two-parameter characterization covering all $q_1,q_2$ simultaneously.
  • The factorization $C_{\mu,\beta}=D^\beta F_\mu*C_{\beta-1}$ is reusable beyond this paper: operators built as Hadamard products with functions whose fractional derivatives lie in mixed norm spaces will obey similar boundedness dichotomies, with logarithmic factors appearing at endpoint cases such as $\gamma_2=\gamma_1+\beta$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies generalized Cesàro-type operators C_{\mu,\beta}, where \mu is a positive Borel measure on [0,1) and \beta>0, defined via the moments \mu_n = \int_0^1 t^n d\mu(t) and the integral representation C_{\mu,\beta} f(z) = \int_0^1 f(tz)/(1-tz)^\beta d\mu(t). The main result (Theorem 6.10) characterizes boundedness from H(p,q,\gamma_1) into H(p,q,\gamma_2) for any 1\le p<\infty, 0<q<\infty, with \gamma_2<\gamma_1+\beta, by the condition that \mu is a (\beta+\gamma_1-\gamma_2)-Carleson measure; endpoint cases p=\infty and q=\infty are also included. Section 7 characterizes boundedness from H(p,\infty,\gamma) into H(p,q,\gamma) for q<\infty via membership of a fractional derivative of F_\mu in the target space (Theorem 7.5). The paper recovers several known results on Hardy, weighted Bergman, and mixed norm spaces as corollaries. The proofs are based on the factorization C_{\mu,\beta} f = F_\mu * (f K_{\beta-1}) and on Carleson-measure descriptions of fractional derivatives of F_\mu.

Significance. If the results hold, they provide a sharp and unifying description of boundedness of Cesàro-type operators on the mixed norm scale; the Carleson exponent s = \beta+\gamma_1-\gamma_2 is exactly the right quantity, and the equivalence between boundedness for one pair (p,q) and for all pairs (p,q) is a strong and useful conclusion. The proofs are detailed and self-contained, with complete estimates for the key lemmas (3.4, 6.8, 6.15) and theorems (6.10, 6.16, 6.17, 7.5). The paper also gives applications to weighted Bergman spaces and recovers prior results, including [13, Theorem 2], as corollaries. The main gap is the unstated finiteness of \mu in the definition of the operator; once that is corrected, the central claims are sound.

major comments (1)
  1. [Section 1, Definition 1.1 and Definition 4.1] The paper defines C_{\mu,\beta} and F_\mu for an arbitrary positive Borel measure \mu on [0,1), but the moments \mu_n = \int_0^1 t^n d\mu(t) need not be finite unless \mu is finite. If \mu([0,1)) = \infty, then \mu_0 = \infty and the constant function 1 is not mapped to a finite analytic function, contradicting the statement immediately after (1.1) that C_{\mu,\beta}(f) \in H(D) for every f \in H(D). This is not a cosmetic issue: all theorems in Sections 6 and 7 are phrased for 'a positive Borel measure' without qualification, yet boundedness is only meaningful when the operator is well-defined. The s-Carleson conditions in the theorems force \mu to be finite a posteriori, so the internal proofs are sound for finite measures; however, the standing assumption should be explicitly stated. I recommend adding 'finite positive Borel measure' (or 'positive Borel measure with finite moments and convergent integral representation') to Definition 1.1, Definition 4.1, and to the hypotheses of the main theorems.
minor comments (5)
  1. [References] In reference [4], the page range '44–644' appears to be a typo; the article in Canad. J. Math. 47 (1995) is on pages 44–64.
  2. [Section 3, proof of Lemma 3.4] The displayed estimate '2rM_q^p(f,r)' uses a notation M_q^p that is not introduced; it should denote the q-th power of the integral mean M_p(f,r). Please clarify the notation.
  3. [Section 6, Lemma 6.9 and Section 7] The function P^*(f)(z) = \sup_{0<t<1} |f(tz)| is called the Poisson maximal function, but it is the radial maximal function; renaming it would avoid conflict with the standard Poisson maximal function.
  4. [Section 4, proof of Lemma 4.3] The convolution formula is written as K_\alpha(e^{i\theta} z) f(e^{-i\theta}); the standard formula is f*g(z) = \int f(e^{i\theta}) g(z e^{-i\theta}) d\theta/(2\pi). The displayed form is correct up to a change of variables but is confusing; please use the standard form.
  5. [Abstract and Introduction] The notation '0<p,q\le\infty' in the abstract suggests that the main results cover p<1, but the main theorems (Theorem 6.10 and Theorem 6.16) are stated for p\ge 1; this should be clarified to avoid overstating the range.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 6.10 is derived from Carleson-measure characterizations and explicit test functions, and the self-cited results are used only as technical tools or as corollaries.

full rationale

The central equivalence (Theorem 6.10) does not reduce to its inputs. The necessity direction (boundedness implies Carleson) is proved in Lemma 6.8 by applying the operator to the explicit test function f_r(z) = (1-rz)^{-(1/p1+1/q1+gamma1)} and integrating against the Fejer-Riesz inequality; the relevant constants are computed directly from the definition of the mixed norm, not from the desired Carleson conclusion. The sufficiency direction (Carleson implies boundedness) uses Proposition 5.1 and Theorem 5.2, both proved in the paper from (5.3) and Proposition 4.4, and the Hadamard-product estimate in Lemma 2.2; the hypothesis on mu is exactly the s-Carleson condition, and the conclusion C_{mu,beta}(H(p,q,gamma1)) subset H(p,q,gamma2) follows with s = beta + gamma1 - gamma2 without any norm being renamed. The results from the authors' earlier papers [4,5,6] are cited as prior published theorems; [4, Theorem A] (Lemma 3.4) is even reproved in the text, and [4, Lemma 2.1] and [5,6] are used only as external technical facts or as results later recovered as corollaries, not as premises that force Theorem 6.10. Corollary 6.13 recovers [13, Theorem 2] from Theorem 6.10 rather than using it as input, so the derivation is not a renaming of a known result. The only substantive caveat found is a well-definedness formulation gap: Definition (1.1) does not explicitly state that the positive Borel measure mu is finite, although the moments mu_n = integral_0^1 t^n dmu(t) and the integral defining C_{mu,beta} may diverge for infinite measures; this is a correctness and assumption issue, not circularity, and the boundedness hypotheses in Lemma 6.8 and Theorem 6.10 implicitly force finiteness. No fitted parameter is called a prediction, no uniqueness theorem is imported from the authors, and no ansatz is smuggled in via citation. The paper is therefore self-contained in the sense relevant to circularity, and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard theorems in analysis: Hardy-Littlewood inclusions, Carleson measure characterizations, radial maximal function estimates, the Fejer-Riesz inequality, and the previously established boundedness of the weighted Cesàro operator. The paper also uses sequence-space characterizations from Blasco's earlier work [4] and moment characterizations of Carleson measures from Chatzifountas-Girela-Pelaez [8]. These are external benchmarks, not restatements of the central claim. No ad hoc parameters or invented entities are introduced.

assumptions (6)
  • standard math Hardy-Littlewood embedding H^p ⊂ H(q,p,1/p - 1/q) (Duren, Theorem 5.11).
    Used to derive mixed norm inclusions (2.5)-(2.7) that many estimates rely on, e.g., in Theorems 6.5 and 6.16.
  • domain assumption The weighted Cesàro operator C_beta-1 is bounded on H(p,q,gamma) for 0<p,q<infinity (Andersen [1]).
    Treated as a black box in Lemma 6.4 and Theorems 6.5 and 6.17; the paper improves it for p>=1 but does not reprove the base case.
  • standard math Radial maximal function estimate M_p(P*f, r) ≲ M_p(f, r) for p>=1.
    Used in Lemma 6.9 and Lemma 6.15 to bound integral means of the maximal function appearing in the pointwise estimates.
  • domain assumption Moment characterization of s-Carleson measures: mu_n = O((n+1)^{-s}) (Chatzifountas-Girela-Pelaez [8, Proposition 1]).
    Used to convert between geometric Carleson conditions and moment conditions in Propositions 4.4 and 5.1 and Theorem 5.2.
  • standard math Fejer-Riesz inequality (Duren [9, Theorem 3.13]).
    Used in Lemma 6.8 to lower bound the H(p,q,gamma) norm of the test functions via boundary integrals.
  • domain assumption Kellogg space characterizations of H(p,q,gamma) norms by coefficient blocks (Blasco [4]).
    Used in Theorem 4.5 and Theorem 7.5 to translate membership of D^alpha F_mu in H(p,q,gamma) into explicit ell^q conditions on the moments mu_n. This is a self-cited but previously published and independent tool.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ces\`aro-type operators on mixed norm spaces." pith.science (2026). https://pith.science/paper/CUP7TEK4

@misc{pith2026250720586,
  author       = {Pith},
  title        = {Pith review of: Ces\`aro-type operators on mixed norm spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUP7TEK4}},
  note         = {Machine review of arXiv:2507.20586}
}
abstract

Given a positive Borel measure $\mu$ on $[0,1)$ and a parameter $\beta>0$, we consider the Ces\`aro-type operator $\mathcal C_{\mu,\beta}$ acting on the analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ on the unit disc of the complex plane $\mathbb D$, defined by \[ \mathcal C_{\mu,\beta}(f)(z)= \sum_{n=0}^\infty \mu_n \left( \sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{(n-k)! \Gamma(\beta)} a_k \right) z^n = \int_0^1 \frac{f(tz)}{(1-tz)^\beta} d\mu(t), \] where $\mu_n=\int_0^1 t^n d\mu(t)$. This operator generalizes the classical Ces\`aro operator (corresponding to the case where $\mu$ is the Lebesgue measure and $\beta=1$) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of $\mathcal C_{\mu,\beta}$ on mixed norm spaces $H(p,q,\gamma)$ for $0<p,q\leq\infty$ and $\gamma>0$. Our results extend and unify several known characterizations for the boundedness of Ces\`aro-type operators acting on spaces of analytic functions.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    K. F. Andersen, Ces` aro averaging operators on Hardy spaces, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 617–624

    work page 1996

  2. [2]

    G. Bao, F. Sun, H. Wulan, Carleson measures and the range of a Ces` aro-like operator acting on H ∞, Anal. Math. Phys. 12 (2022) 142

    work page 2022

  3. [3]

    G. Bao, K. Guo, F. Sun, H. Wulan, Hankel matrices acting on the Dirichlet space J. Fourier Anal. Appl. 30 (2024) Paper No. 53, 26 pp

    work page 2024

  4. [4]

    Blasco, Multipliers on spaces of analytic functions, Canad

    O. Blasco, Multipliers on spaces of analytic functions, Canad. J. Math. 47 (1995), no. 1, 44–644

    work page 1995

  5. [5]

    Blasco, Ces` aro-type operators on Hardy spaces,J

    O. Blasco, Ces` aro-type operators on Hardy spaces,J. Math. Anal. Appl. 529 (2024), no. 2, Paper No. 127017, 26 pp

    work page 2024

  6. [6]

    Blasco, Generalized Ces` aro operators on weighted Dirichlet spaces,J

    O. Blasco, Generalized Ces` aro operators on weighted Dirichlet spaces,J. Math. Anal. Appl. 540 (2024), Paper No. 128627, 21 pp

    work page 2024

  7. [7]

    Blasco and H

    O. Blasco and H. Jarchow, A note on Carleson measures for Hardy spaces, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 371–389

    work page 2005

  8. [8]

    Chatzifountas, D

    C. Chatzifountas, D. Girela ´Alvarez and J. ´A. Pel´ aez, A generalized Hilbert matrix acting on Hardy spaces, J. Math. Anal. Appl. 413 (2014), no. 1, 154–168

    work page 2014

Show all 29 references
  1. [9]

    P. L. Duren, Theory of Hp-spaces, Pure and Applied Mathematics, Vol. 38 (1970). Aca- demic Press, New York-London

    1970

  2. [10]

    Galanopoulos et al., Operators induced by radial measures acting on the Dirichlet space, Results Math

    P. Galanopoulos et al., Operators induced by radial measures acting on the Dirichlet space, Results Math. 78 (2023), no. 3, Paper No. 106, 24 pp

    2023

  3. [11]

    Galanopoulos, D

    P. Galanopoulos, D. Girela and N. Merch´ an, Ces` aro-like operators acting on spaces of analytic functions, Anal. Math. Phys. 12 (2022), no. 2, Paper No. 51, 29 pp

    2022

  4. [12]

    Galanopoulos, D

    P. Galanopoulos, D. Girela and N. Merch´ an, Ces` aro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions, J. Math. Anal. Appl. 526 (2023), no. 2, Paper No. 127287, 13 pp

    2023

  5. [13]

    Galanopoulos, A

    P. Galanopoulos, A. G. Siskakis and R. Zhao, Weighted Ces` aro type operators between weighted Bergman spaces, Bull. Sci. Math. 202 (2025), Paper No. 103622, 17 pp

    2025

  6. [14]

    Y. T. Guo, P. C. Tang and X. J. Zhang, Ces` aro-like operators between the Bloch space and Bergman spaces, Ann. Funct. Anal. 15 (2024), no. 1, Paper No. 8, 16 pp

    2024

  7. [15]

    Hardy, Notes on some points in the integral calculus LXVI: the arithmetic mean of a Fourier constant, Messenger Math

    G.H. Hardy, Notes on some points in the integral calculus LXVI: the arithmetic mean of a Fourier constant, Messenger Math. 58 (1929) 50–52

    1929

  8. [16]

    Hardy, J.E

    G.H. Hardy, J.E. Littlewood, Some new properties of Fourier constants, J. Lond. Math. Soc. 6 (1931) 3–9

    1931

  9. [17]

    Hedenmalm, B

    H. Hedenmalm, B. Korenblum, and K. Zhu Theory of Bergman Spaces volume 199 of Graduate Texts in Mathematics (2000). Springer-Verlag, New York

    2000

  10. [18]

    J. J. Jin and S. A. Tang, Generalized Ces` aro operators on Dirichlet-type spaces,Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), no. 1, 212–220

    2022

  11. [19]

    C. N. Kellogg, An extension of the Hausdorff-Young theorem, Michigan Math. J. 18 (1971), 121–127

    1971

  12. [20]

    Lin and H

    Q. Lin and H. Xie, Ces` aro-type operators on derivative-type Hilbert spaces of anlytic func- tions: the proof of a conjecture, J. Funct. Anal. 288 (2025), Paper No. 1110813, 22 pp

    2025

  13. [21]

    Miao, The Ces` aro operator is bounded on Hp for 0 < p <1, Proc

    J. Miao, The Ces` aro operator is bounded on Hp for 0 < p <1, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1077–1079

    1992

  14. [22]

    Pavlovic, Analytic functions with decreasing coefficients and Hardy and Bloch spaces, Proc

    M. Pavlovic, Analytic functions with decreasing coefficients and Hardy and Bloch spaces, Proc. Edinb. Math. Soc. 56 (2013) 623–625

    2013

  15. [23]

    Rhaly, Terraced matrices, Bull

    H. Rhaly, Terraced matrices, Bull. London Math. Soc. 21 (1989) 399–406. 24 OSCAR BLASCO AND ALEJANDRO MAS

    1989

  16. [24]

    Rhaly, p-Ces` aro matrices,Houston J

    H. Rhaly, p-Ces` aro matrices,Houston J. Math. 15 (1989) 137-146

    1989

  17. [25]

    A. G. Siskakis, Composition semigroups and the Ces` aro operator onHp, J. London Math. Soc. (2) 36 (1987), no. 1, 153–164

    1987

  18. [26]

    A. G. Siskakis, On the Bergman space norm of the Ces` aro operator, Arch. Math. (Basel) 67 (1996), no. 4, 312–318

    1996

  19. [27]

    A. G. Siskakis, The Ces` aro operator is bounded onH 1, Proc. Amer. Math. Soc. 110 (1990), no. 2, 461–462

    1990

  20. [28]

    Stempak, Ces` aro averaging operators, Proc

    K. Stempak, Ces` aro averaging operators, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 1, 121–126

    1994

  21. [29]

    Zhao and K

    R. Zhao and K. Zhu, Theory of Bergman spaces in the unit ball of Cn, M´ em. Soc. Math. Fr. (N.S.) (2008), no. 115, vi+103 pp. Departamento de An´alisis Matem´atico, Universitat de Val`encia, Burjassot 46100, Valencia (Spain) Email address: oscar.blasco@uv.es Departamento de Ma...

    2008

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.