Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliers

Alberto Arenas; Edgar Labarga; \'Oscar Ciaurri

arxiv: 1906.07999 · v1 · pith:WSSS5NSAnew · submitted 2019-06-19 · 🧮 math.CA

Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein g_(k)-functions and the Laplace type multipliers

Alberto Arenas , \'Oscar Ciaurri , Edgar Labarga This is my paper

Pith reviewed 2026-05-25 20:07 UTC · model grok-4.3

classification 🧮 math.CA

keywords Littlewood-Paley-Stein functionsJacobi expansionsnorm equivalencesweighted spacesLaplace multipliersorthogonal polynomialsharmonic analysisthree-term recurrence

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The pith

Littlewood-Paley-Stein g_k functions tied to the Jacobi recurrence operator satisfy weighted norm equivalences that imply results for Laplace-type multipliers.

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

This paper continues the study of harmonic analysis for Jacobi expansions by introducing the operator J^(α,β) equal to the three-term recurrence operator minus the identity. It defines the associated Littlewood-Paley-Stein g_k^(α,β)-functions and proves that these functions are equivalent in weighted norms to the original functions. The equivalence then produces a boundedness result for multipliers of Laplace type on the corresponding weighted spaces.

Core claim

Given the operator J^(α,β) = J^(α,β) - I, where J^(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials, the corresponding Littlewood-Paley-Stein g_k^(α,β)-functions satisfy an equivalence of norms with weights, from which a result for Laplace type multipliers follows.

What carries the argument

The Littlewood-Paley-Stein g_k^(α,β)-functions constructed from the operator J^(α,β) - I that encodes the three-term recurrence of normalized Jacobi polynomials.

If this is right

The g_k functions characterize the weighted L^p spaces associated with the Jacobi expansions.
Laplace-type multipliers are bounded on the weighted L^p spaces once the norm equivalence is available.
The results apply uniformly to the family of normalized Jacobi polynomials for the given range of α and β.

Where Pith is reading between the lines

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

The same recurrence-based construction of g-functions could be tested on other classical orthogonal polynomial systems that obey three-term recurrences.
The multiplier theorem might extend to discrete operators defined by similar second-order difference equations beyond the Jacobi case.
Connections could be explored between these discrete g-functions and their continuous counterparts on the interval with Jacobi weight.

Load-bearing premise

The three-term recurrence structure of the normalized Jacobi polynomials together with the weighted estimates from the two prior papers in the series continue to hold for the parameters and weight classes under consideration.

What would settle it

A concrete pair of α, β and a weight in the admissible class for which the L^p norm of the g_k function differs from the L^p norm of the function by an arbitrarily large factor.

read the original abstract

The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where $J^{(\alpha,\beta)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood-Paley-Stein $g_k^{(\alpha,\beta)}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

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

This paper adds the g_k functions for J^(α,β)-I and derives the weighted equivalence plus multiplier result as a direct sequel to the authors' two prior installments.

read the letter

The punchline is that this paper defines the g_k^(α,β) functions for the operator J^(α,β) - I on normalized Jacobi polynomials and establishes their weighted norm equivalence, from which a Laplace multiplier theorem follows. It is a direct continuation of the two earlier papers by the same group. The new content is the specific square functions and the multiplier deduction. The paper does what it sets out to do by linking the g-functions to the recurrence relation and carrying over the weighted estimates. That gives a clean characterization. The soft spot is the reliance on ACL-JacI and ACL-JacII. The equivalence is presented as following from the three-term recurrence and those prior results, but the transfer to the exact g_k construction and chosen weights needs to hold for the full range of α, β. If it does, the argument is fine; if the priors have gaps for these parameters, the multiplier result weakens. This paper is for specialists tracking the Jacobi expansion harmonic analysis program. A general reader in harmonic analysis would find it too narrow without the background. I recommend sending it for peer review. The series is building a coherent body of work, and the claims are precise enough to be checked.

Referee Report

1 major / 1 minor

Summary. The paper continues the authors' series on discrete harmonic analysis for Jacobi expansions. It defines Littlewood-Paley-Stein g_k^(α,β)-functions via the operator J^(α,β) = J^(α,β) - I (the three-term recurrence operator on normalized Jacobi polynomials minus the identity) and proves weighted norm equivalences for these functions; as a consequence it obtains a result on Laplace-type multipliers.

Significance. If the weighted norm equivalences are established under the stated hypotheses, the work supplies a discrete Littlewood-Paley theory adapted to Jacobi expansions that yields multiplier bounds; this is a natural and potentially useful extension of the square-function machinery developed in the preceding papers of the series.

major comments (1)

[Statement of the main theorem and its proof (following the abstract)] The central claim that the g_k^(α,β)-functions satisfy weighted norm equivalence (and therefore yield the multiplier result) rests on an implicit transfer of maximal-function and square-function estimates from ACL-JacI and ACL-JacII. The manuscript does not contain an explicit verification that the chosen ranges of α, β and the Muckenhoupt-type weight classes employed here lie inside the hypotheses already proved in those references; this verification is load-bearing for both the equivalence and the multiplier conclusion.

minor comments (1)

[Abstract] The abstract states that the equivalence holds 'with weights' but does not name the precise weight class (e.g., A_p^α,β or similar); this should be stated explicitly already in the abstract and introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of parameter ranges. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim that the g_k^(α,β)-functions satisfy weighted norm equivalence (and therefore yield the multiplier result) rests on an implicit transfer of maximal-function and square-function estimates from ACL-JacI and ACL-JacII. The manuscript does not contain an explicit verification that the chosen ranges of α, β and the Muckenhoupt-type weight classes employed here lie inside the hypotheses already proved in those references; this verification is load-bearing for both the equivalence and the multiplier conclusion.

Authors: We agree that an explicit verification strengthens the presentation. The ranges of α, β (typically α, β > -1) and the Muckenhoupt weights A_p^α,β are identical to those under which the maximal and square-function bounds were established in ACL-JacI and ACL-JacII. In the revised manuscript we will insert a short paragraph (likely after the statement of the main theorem) that recalls the precise hypotheses from the earlier papers and confirms that our choices lie strictly inside those hypotheses, thereby justifying the transfer of the estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new equivalence proved using recurrence and prior background

full rationale

The paper continues a series but defines the g_k functions from the operator J^(α,β)-I and states that it proves the weighted norm equivalence directly in this work via the three-term recurrence structure. Citations to ACL-JacI and ACL-JacII supply background properties on maximal functions or weighted inequalities, which are independent prior results rather than reductions of the present claims. No step reduces a prediction or central result to a fit, self-definition, or unverified self-citation chain by construction. This is a standard continuation paper with normal self-citation that does not make the derivation circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work relies on the three-term recurrence for normalized Jacobi polynomials (standard in the field) and on results from the two cited earlier papers by the same authors.

axioms (2)

domain assumption The three-term recurrence relation for normalized Jacobi polynomials defines a bounded self-adjoint operator J^(α,β) on the appropriate weighted L2 space.
Invoked in the definition of the operator J^(α,β) = J^(α,β) - I.
domain assumption Results established in ACL-JacI and ACL-JacII remain valid for the parameter ranges and weight classes used here.
The abstract states that the present work continues the research carried out in those two papers.

pith-pipeline@v0.9.0 · 5648 in / 1493 out tokens · 23127 ms · 2026-05-25T20:07:54.914861+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Arenas, ´O

A. Arenas, ´O. Ciaurri, and E. Labarga, Discrete harmonic analysis asso ciated with Jacobi expansions I: the heat semigroup, preprint, arXiv: 1806.00 056 (2018)

work page 2018
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A weighted transplantation theorem for Jacobi coefficients

A. Arenas, ´O. Ciaurri, and E. Labarga, A weighted transplantation theo rem for Jacobi coef- ﬁcients, preprint, arXiv:1812.08422 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
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Arenas, ´O

A. Arenas, ´O. Ciaurri, and E. Labarga, Discrete harmonic analysis asso ciated with Jacobi expansions II: the Riesz transform, preprint

work page
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Askey, A transplantation theorem for Jacobi coeﬃcien ts, Paciﬁc J

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work page 1967
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J. J. Betancor, A. J. Castro, J. C. Fari˜ na, and L. Rodr ´ ıguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arX iv: 1512.01379 (2015)

work page arXiv 2015
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J. J. Betancor, A. J. Castro, and A. Nowak, Calder´ on-Zyg mund operators in the Bessel setting, Monatsh. Math. 167 (2012), 375–403

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Ciaurri, T

´O. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea, and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017), 109–131

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Ciaurri and L

´O. Ciaurri and L. Roncal, Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions, J. Funct. Anal. 258 (2010), 2173–2204

work page 2010
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Garrig´ os, E

G. Garrig´ os, E. Harboure, T. Signes, J. L. Torrea, and B. Viviani, A sharp weighted trans- plantation theorem for Laguerre function expansions, J. Funct. Anal. 244 (2007), 247–276

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J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II , Springer-Verlag, Berlin, 1979

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J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourie r series and power series (II), Proc. London Math. Soc. 42 (1936), 52–89

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Nowak, P

A. Nowak, P. Sj¨ ogren, and T. S. Szarek, Analysis relate d to all admissible type parameters in the Jacobi setting, Constr. Approx. 41 (2015), 185–218

work page 2015
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Nowak, K

A. Nowak, K. Stempak, and T. Z. Szarek, On harmonic analy sis operators in Laguerre-Dunkl and Laguerre-symmetrized settings, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 096, 39 pp

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E. M. Stein, On the functions of Littlewood-Paley, Lusi n, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958) 430–466

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E. M. Stein, Topics in harmonic analysis related to the Littlewood-Pale y theory , Annals of Mathematics Studies, No. 63 Princeton University Press, Pr inceton, N.J., 1970

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Stempak and J

K. Stempak and J. L. Torrea, On g-functions for Hermite function expansions, Acta Math. Hungar. 109 (2005), 99–125

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Szeg˝ o, Orthogonal polynomials , American Mathematical Society Providence, Rhode Is- land, 1975

G. Szeg˝ o, Orthogonal polynomials , American Mathematical Society Providence, Rhode Is- land, 1975. 16 A. ARENAS, ´O. CIAURRI, AND E. LABARGA Departamento de Matem ´aticas y Computaci ´on, Universidad de La Rioja, Complejo Cient´ıfico-Tecnol´ogico, Calle Madre de Dios 53, 26006 Logro ˜no, Spain E-mail address : alberto.arenas@unirioja.es Departamento de ...

work page 1975

[1] [1]

Arenas, ´O

A. Arenas, ´O. Ciaurri, and E. Labarga, Discrete harmonic analysis asso ciated with Jacobi expansions I: the heat semigroup, preprint, arXiv: 1806.00 056 (2018)

work page 2018

[2] [2]

A weighted transplantation theorem for Jacobi coefficients

A. Arenas, ´O. Ciaurri, and E. Labarga, A weighted transplantation theo rem for Jacobi coef- ﬁcients, preprint, arXiv:1812.08422 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[3] [3]

Arenas, ´O

A. Arenas, ´O. Ciaurri, and E. Labarga, Discrete harmonic analysis asso ciated with Jacobi expansions II: the Riesz transform, preprint

work page

[4] [4]

Askey, A transplantation theorem for Jacobi coeﬃcien ts, Paciﬁc J

R. Askey, A transplantation theorem for Jacobi coeﬃcien ts, Paciﬁc J. Math. 21 (1967), 393–404

work page 1967

[5] [5]

J. J. Betancor, A. J. Castro, J. C. Fari˜ na, and L. Rodr ´ ıguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arX iv: 1512.01379 (2015)

work page arXiv 2015

[6] [6]

J. J. Betancor, A. J. Castro, and A. Nowak, Calder´ on-Zyg mund operators in the Bessel setting, Monatsh. Math. 167 (2012), 375–403

work page 2012

[7] [7]

Ciaurri, T

´O. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea, and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017), 109–131

work page 2017

[8] [8]

Ciaurri and L

´O. Ciaurri and L. Roncal, Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions, J. Funct. Anal. 258 (2010), 2173–2204

work page 2010

[9] [9]

Garrig´ os, E

G. Garrig´ os, E. Harboure, T. Signes, J. L. Torrea, and B. Viviani, A sharp weighted trans- plantation theorem for Laguerre function expansions, J. Funct. Anal. 244 (2007), 247–276

work page 2007

[10] [10]

N. N. Lebedev, Special functions and its applications , Dover, New York, 1972

work page 1972

[11] [11]

Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II , Springer-Verlag, Berlin, 1979

work page 1979

[12] [12]

J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourie r series and power series (II), Proc. London Math. Soc. 42 (1936), 52–89

work page 1936

[13] [13]

Nowak, P

A. Nowak, P. Sj¨ ogren, and T. S. Szarek, Analysis relate d to all admissible type parameters in the Jacobi setting, Constr. Approx. 41 (2015), 185–218

work page 2015

[14] [14]

Nowak, K

A. Nowak, K. Stempak, and T. Z. Szarek, On harmonic analy sis operators in Laguerre-Dunkl and Laguerre-symmetrized settings, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 096, 39 pp

work page 2016

[15] [15]

F. W. J. Olver (editor-in-chief), NIST Handbook of Mathematical Functions , Cambridge University Press, New York, 2010

work page 2010

[16] [16]

A. P. Prudnikov, A. Y. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1. Elementary functions, Gordon and Breach Science Publishers, New York, 1986

work page 1986

[17] [17]

E. M. Stein, On the functions of Littlewood-Paley, Lusi n, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958) 430–466

work page 1958

[18] [18]

E. M. Stein, Topics in harmonic analysis related to the Littlewood-Pale y theory , Annals of Mathematics Studies, No. 63 Princeton University Press, Pr inceton, N.J., 1970

work page 1970

[19] [19]

Stempak and J

K. Stempak and J. L. Torrea, On g-functions for Hermite function expansions, Acta Math. Hungar. 109 (2005), 99–125

work page 2005

[20] [20]

Szeg˝ o, Orthogonal polynomials , American Mathematical Society Providence, Rhode Is- land, 1975

G. Szeg˝ o, Orthogonal polynomials , American Mathematical Society Providence, Rhode Is- land, 1975. 16 A. ARENAS, ´O. CIAURRI, AND E. LABARGA Departamento de Matem ´aticas y Computaci ´on, Universidad de La Rioja, Complejo Cient´ıfico-Tecnol´ogico, Calle Madre de Dios 53, 26006 Logro ˜no, Spain E-mail address : alberto.arenas@unirioja.es Departamento de ...

work page 1975