Vector-valued $q$-variational inequalities for averaging operators and Hilbert transform

Guixiang Hong; Tao Ma; Wei Liu

arxiv: 1907.11633 · v1 · pith:FBXVGKMJnew · submitted 2019-07-26 · 🧮 math.CA

Vector-valued q-variational inequalities for averaging operators and Hilbert transform

Guixiang Hong , Wei Liu , Tao Ma This is my paper

Pith reviewed 2026-05-24 15:08 UTC · model grok-4.3

classification 🧮 math.CA

keywords vector-valued variational inequalitiesmartingale cotype qUMD propertyHilbert transformaveraging operatorsBanach spacesL^p boundedness

0 comments

The pith

Martingale cotype q is necessary for vector-valued q-variational inequalities of averaging operators, with UMD and cotype q characterized via those for the Hilbert transform.

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

The paper proves that the martingale cotype q property of a Banach space is required for the L^p boundedness of vector-valued q-variational inequalities for averaging operators. This necessity closes a question left open by earlier sufficiency results that held only in spaces already known to have the cotype property. The authors further show that the same style of inequalities for the Hilbert transform can be used to characterize both the UMD property and the martingale cotype q property. These characterizations tie the geometric features of the target Banach space directly to the size of the variational expressions.

Core claim

Martingale cotype q is necessary for the boundedness of vector-valued q-variational inequalities for averaging operators, and the UMD property together with martingale cotype q can be characterized by the corresponding inequalities for the Hilbert transform.

What carries the argument

Vector-valued q-variational inequalities for averaging operators and the Hilbert transform, which measure the size of the q-variation of the operator applied to vector-valued functions.

If this is right

Any Banach space in which the averaging-operator inequalities hold must possess martingale cotype q.
The Hilbert-transform inequalities serve as a test for both UMD and martingale cotype q.
Failure of the inequalities in a given space implies failure of the corresponding geometric property.
The results apply uniformly across the range of p where the inequalities are considered.

Where Pith is reading between the lines

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

The necessity result may extend to other singular integral operators beyond averaging and Hilbert.
One could test the geometric properties of concrete function spaces by checking whether the variational inequalities hold for them.
The characterizations suggest that variational inequalities could serve as an alternative definition of UMD and cotype q in some contexts.

Load-bearing premise

The definitions of the vector-valued q-variational inequalities and the earlier sufficiency theorems for spaces with martingale cotype q are taken as given.

What would settle it

A Banach space lacking martingale cotype q for which the vector-valued q-variational inequality for averaging operators remains bounded on L^p would disprove the necessity claim.

read the original abstract

Recently, in \cite{GXHTM}, the authors established $L^p$-boundedness of vector-valued $q$-variational inequalities for averaging operators which take values in the Banach space satisfying martingale cotype $q$ property. In this paper, we prove that martingale cotype $q$ property is also necessary for the vector-valued $q$-variational inequalities, which is a question left open. Moreover, we characterize UMD property and martingale cotype $q$ property in terms of vector valued $q$-variational inequalities for Hilbert transform.

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 closes the necessity gap left open for averaging operators and adds Hilbert-transform characterizations, but everything rides on matching the prior paper's exact setup.

read the letter

The main takeaway is that this paper supplies the missing necessity direction: martingale cotype q is required for the vector-valued q-variational inequalities to hold for averaging operators. It also gives operator characterizations of both the UMD property and martingale cotype q via the Hilbert transform. That necessity piece was explicitly left open in the cited earlier work, so the paper finishes that specific loop. The Hilbert-transform characterizations look like fresh content built on the same framework. The work is technically grounded in the sense that it directly engages the open question and uses contraposition from the known sufficiency result. Credit is due for completing the picture without introducing new free parameters or invented objects. The soft spot is that the necessity claim stands or falls with precise agreement on the q-variation seminorm, the range of p and q, and the formulation of the averaging operators from the prior paper. Any small mismatch in those definitions would break the implication, and the abstract gives no room to verify the matching. The Hilbert-transform part inherits the same dependence. This is narrow-gauge work aimed at people already inside vector-valued harmonic analysis who care about Banach-space properties and variational inequalities. A reader outside that subfield will not get much from it. It is coherent on its own terms and engages the literature honestly, so it deserves a serious referee even if the proofs turn out to need tightening on the definition-matching step. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the martingale cotype q property of a Banach space is necessary for the L^p-boundedness of vector-valued q-variational inequalities associated to averaging operators, resolving an open question left in the sufficiency result of [GXHTM]. It further establishes characterizations of both the UMD property and the martingale cotype q property via the boundedness of the corresponding vector-valued q-variational inequalities for the Hilbert transform.

Significance. If the necessity and characterization results hold with matching definitions and ranges, the paper completes the equivalence between the analytic inequalities and the geometric properties of the target Banach space. This supplies the missing necessity direction for averaging operators and yields new characterizations for the Hilbert transform, both of which are central to vector-valued harmonic analysis.

major comments (2)

[Section 2 / necessity argument] The necessity argument for averaging operators proceeds by contraposition from the L^p-boundedness established in [GXHTM]. Section 2 (or the relevant necessity section) must therefore verify that the precise definition of the q-variation seminorm, the admissible range of p and q, and the formulation of the averaging operators coincide exactly with those used in [GXHTM]; any discrepancy would invalidate the implication that boundedness fails whenever martingale cotype q fails.
[Section 3 / Hilbert-transform characterizations] For the Hilbert-transform characterizations, the paper must confirm that the vector-valued q-variational inequality is stated with the same parameters (including the precise range of q relative to the cotype and the UMD constant) that appear in the averaging-operator case, so that the two characterizations are consistent with each other and with the necessity result.

minor comments (2)

[Notation section] Notation for the q-variation seminorm should be introduced once and used uniformly; the current alternation between V_q and the explicit supremum expression is distracting.
[Theorem 1.1 / main necessity theorem] The statement of the main necessity theorem should explicitly list the range of p and q for which the implication holds, rather than referring the reader solely to [GXHTM].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for identifying the need to make definitional consistency explicit. We address both major comments below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section 2 / necessity argument] The necessity argument for averaging operators proceeds by contraposition from the L^p-boundedness established in [GXHTM]. Section 2 (or the relevant necessity section) must therefore verify that the precise definition of the q-variation seminorm, the admissible range of p and q, and the formulation of the averaging operators coincide exactly with those used in [GXHTM]; any discrepancy would invalidate the implication that boundedness fails whenever martingale cotype q fails.

Authors: We confirm that the q-variation seminorm (defined via the usual sup over partitions), the admissible ranges (1 < p < ∞ and q > 2), and the averaging operators (dyadic or continuous, as appropriate) are identical to those in [GXHTM]. This is built into the setup of our necessity argument. To address the referee's request explicitly, we will insert a short paragraph at the start of Section 2 stating that all notions and parameter ranges coincide exactly with those of [GXHTM]. revision: yes
Referee: [Section 3 / Hilbert-transform characterizations] For the Hilbert-transform characterizations, the paper must confirm that the vector-valued q-variational inequality is stated with the same parameters (including the precise range of q relative to the cotype and the UMD constant) that appear in the averaging-operator case, so that the two characterizations are consistent with each other and with the necessity result.

Authors: The q-variational inequalities for the Hilbert transform are formulated with exactly the same parameters as in the averaging-operator setting: the same range of q (q > 2 when the space has martingale cotype q, and the UMD constant appearing in the same way), the same p-range, and the same definition of the q-variation seminorm. This ensures the characterizations are consistent with the necessity result for averaging operators. We will add an explicit sentence in Section 3 (and a cross-reference in the introduction) confirming that the parameters match those used for averaging operators. revision: yes

Circularity Check

0 steps flagged

Necessity and characterization results are independent proofs building on prior sufficiency

full rationale

The paper cites [GXHTM] solely for the established sufficiency direction (L^p boundedness when martingale cotype q holds) and then supplies a separate proof that the property is necessary, plus a new characterization for the Hilbert transform. No equation or claim in the abstract reduces the necessity statement to a re-labeling, re-fitting, or direct contraposition that would be true by definition from the cited sufficiency alone; the definitions are shared for consistency but the necessity implication requires additional argument. This is standard incremental work with one self-citation that is not load-bearing for the new claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior sufficiency result in [GXHTM] and on the standard definitions of martingale cotype q, UMD, averaging operators, Hilbert transform, and q-variational inequalities in the vector-valued setting; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

domain assumption Standard definitions and properties of martingale cotype q and UMD for Banach spaces from prior literature
The necessity and characterization statements presuppose these properties are well-defined and that the sufficiency direction from [GXHTM] holds.

pith-pipeline@v0.9.0 · 5618 in / 1290 out tokens · 30589 ms · 2026-05-24T15:08:56.859584+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

[1]

Bourgain, Some remarks on Banach spaces in which martingale diﬀerence sequences are unconditional, Ark

J. Bourgain, Some remarks on Banach spaces in which martingale diﬀerence sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163-168

work page 1983
[2]

Bourgain, Vector-valued singular integrals and the H1-BMO duality , Probability theory and harmonic analysis, 1-19, Textbooks Pure Appl

J. Bourgain, Vector-valued singular integrals and the H1-BMO duality , Probability theory and harmonic analysis, 1-19, Textbooks Pure Appl. Math. Dekker, New York, 1986

work page 1986
[3]

Bourgain, Pointwise ergodic theorems for arithmetic sets , Publ

J. Bourgain, Pointwise ergodic theorems for arithmetic sets , Publ. Math. IHES. 69 (1989), 5-41

work page 1989
[4]

Bourgain, M

J. Bourgain, M. Mirek, E. M. Stein, B. Wr´ obel, On dimension-free variational inequalities for averaging operators in Rd, Geom. Funct. Anal. 28 (2018), no. 1, 58-99

work page 2018
[5]

Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$

J. Bourgain, M. Mirek, E. M. Stein, B. Wr´ obel, Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in Zd, arXiv:1804.07679 (to appear)

work page internal anchor Pith review Pith/arXiv arXiv
[6]

D. L. Burkholder, A geometric condition that implies the existence of certain singular in- tegrals of Banach-space-valued functions , Conference on harmonic analysis in honor of An- toni Zygmund, Vol. I, II(Chicago, III., 1981), 270-286, Wadswo rth Math. Ser., Wadsworth, Belmont, CA, 1983

work page 1981
[7]

J. T. Campbell, R. L. Jones, K. Reinhold, M. Wierdl, Oscillation and variation for the Hilbert transform, Duke Math. J. 105 (2000), no. 1, 59-83

work page 2000
[8]

J. T. Campbell, R. L. Jones, K. Reinhold, M. Wierdl, Oscillation and variation for singular integrals in higher dimensions , Trans. Amer. Math. Soc. 355 (2003), no. 5, 2115-2137

work page 2003
[9]

Chen, Y, Ding, G

Y. Chen, Y, Ding, G. Hong, H. Liu, Weighted jump and variational inequalities for rough operators, J. Funct. Anal. 274 (2018), no. 8, 2446-2475

work page 2018
[10]

Y, Ding, G. Hong, H. Liu, Jump and variational inequalities for rough operators , J. Fourier Anal. Appl. 23 (2017), no. 3, 679-711

work page 2017
[11]

Y. Do, C. Muscalu, C. Thiele, Variational estimates for paraproducts , Rev. Mat. Iberoam. 28 (2012), no. 3, 8578-78

work page 2012
[12]

A. T. Gillespie, J. L. Torrea, Dimension free estimates for the oscillation of Riesz trans - forms, Isreal J. Math. 141 (2004), 125-144

work page 2004
[13]

Grafakos, Classical and Modern Fourier Analysis , Pearson/Prentice Hall, Upper-Saddle River, 2004

L. Grafakos, Classical and Modern Fourier Analysis , Pearson/Prentice Hall, Upper-Saddle River, 2004

work page 2004
[14]

G. Hong, T. Ma, Vector valued q-variation for ergodic averages and analytic semigroups , J. Math. Anal. Appl. 437 (2016), no. 2, 1084-1100

work page 2016
[15]

G. Hong, T. Ma, Vector valued q-variation for diﬀerential operators and semigroups I , Math. Z. 286 (2017), no. 1-2, 89-120. VECTOR-V ALUED q-V ARIATIONAL INEQUALITIES... 13

work page 2017
[16]

T. P. Hyt¨ onen, Littlewood-Paley-Stein theory for semigroups in UMD space s, Rev. Mat. Iberoam. 23 (2007), no. 3, 973-1009

work page 2007
[17]

T. P. Hyt¨ onen, M. T. Lacey, C. P´ erez,Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), no. 3, 529-540

work page 2013
[18]

T. P. Hyt¨ onen, M. T. Lacey, I. Parissis, A variation norm Carleson theorem for vector- valued Walsh-Fourier series , Rev. Mat. Iberoam. 30 (2014), no. 3, 979-1014

work page 2014
[19]

T. P. Hyt¨ onen, V. N. Jan, M. Veraar, L. Weis, Analysis in Banach spaces. Vol. I. Mar- tingales and Littlewood-Paley theory , Springer, Cham, 2016

work page 2016
[20]

R. L. Jones, R. Kaufman, J. M. Rosenblatt, M. Wierdl, Oscillation in ergodic theory . Ergodic Theory Dynam. Systems 18 (1998), no. 4, 889-935

work page 1998
[21]

R. L. Jones, J. M. Rosenblatt, M. Wierdl, Oscillation inequalities for rectangles , Proc. Am. Math. Soc. 129 1349-1358 (2000)

work page 2000
[22]

R. L. Jones, J. M. Rosenblatt, M. Wierdl, Oscillation in ergodic theory: higher dimensional results, Israel J. Math. 135 (2003), 1-27

work page 2003
[23]

R. L. Jones, R. Kaufman, J. M. Rosenblatt, M. Wierdl, Oscillation and variation for singular integrals in higher dimensions , Trans. Amer. Math. Soc. 355 (2003), no. 5, 2115- 2137.RG

work page 2003
[24]

R. L. Jones, A. Seeger, J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6711-6742

work page 2008
[25]

R. L. Jones, G. Wang, Variation inequalities for the Fej´ er and Poisson kernels , Trans. Amer. Math. Soc. 356 (2004), no. 11, 4493-4518

work page 2004
[26]

Le Merdy, Q

C. Le Merdy, Q. Xu, Strong q -variation inequalities for analytic semigroups , Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2069-2097 (2013)

work page 2012
[27]

Krause, P

B. Krause, P. Zorin-Kranich, Weighted and vector-valued variational estimates for ergo dic averages, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 244-256

work page 2018
[28]

L´ epingle, La variation d′ordre p des semi-martingales , Z

D. L´ epingle, La variation d′ordre p des semi-martingales , Z. Wahrsch. Verw. Gebiete 36 (1976), no. 4, 295-316

work page 1976
[29]

T. Ma, J. L. Torrea, Q. Xu, Weighted variation inequalities for diﬀerential operator s and singular integrals, J. Funct. Anal. 268 (2015), no. 2, 376-416

work page 2015
[30]

T. Ma, J. L. Torrea, Q. Xu, Weighted variation inequalities for diﬀerential operator s and singular integrals in higher dimensions , Sci. China Math. 60 (2017), no. 8, 1419-1442

work page 2017
[31]

Mart ´ ınez, J

T. Mart ´ ınez, J. L. Torrea, Q, Xu, Vector-valued Littlewood-Paley-Stein theory for semi- groups, Adv. Math. 203 (2006), no. 2, 430-475

work page 2006
[32]

Mirek, E

M. Mirek, E. Stein, B. Trojan, ℓp(Zd)-estimates for discrete operators of Radon type: variational estimates , Invent. Math. 209 (2017), no. 3, 665-748

work page 2017
[33]

Mirek, B

M. Mirek, B. Trojan, P. Zorin-Kranich, Variational estimates for averages and truncated singular integrals along the prime numbers , Trans. Amer. Math. Soc. 369 (2017), no. 8, 5403-5423

work page 2017
[34]

Oberlin, A

R. Oberlin, A. Seeger, T. Tao, C. Thiele, J. Wright, A variation norm Carleson theorem , J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 421-464

work page 2012
[35]

Pisier, Martingale with values in uniformly convex spaces , Israel J

G. Pisier, Martingale with values in uniformly convex spaces , Israel J. Math. 20(1975), no. 3-4, 326-350

work page 1975
[36]

Pisier, Q

G. Pisier, Q. Xu, The strong p-variation of martingale and orthogonal series , Probab. Theory Related ﬁelds 77 (1988), no. 4, 497-514

work page 1988
[37]

Xu, Littlewood-Paley theory for functions with values in unifo rmly convex spaces , J

Q. Xu, Littlewood-Paley theory for functions with values in unifo rmly convex spaces , J. Reine Angew. Math. 504 (1998), 195-226. 14 GUIXIANG HONG, WEI LIU, AND TAO MA School of Mathematics and Statistics, Wuhan University, Wu han 430072 and Hubei Key Laboratory of Computational Science, Wuhan Unive rsity, Wuhan 430072, China E-mail address : guixiang.hon...

work page 1998

[1] [1]

Bourgain, Some remarks on Banach spaces in which martingale diﬀerence sequences are unconditional, Ark

J. Bourgain, Some remarks on Banach spaces in which martingale diﬀerence sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163-168

work page 1983

[2] [2]

Bourgain, Vector-valued singular integrals and the H1-BMO duality , Probability theory and harmonic analysis, 1-19, Textbooks Pure Appl

J. Bourgain, Vector-valued singular integrals and the H1-BMO duality , Probability theory and harmonic analysis, 1-19, Textbooks Pure Appl. Math. Dekker, New York, 1986

work page 1986

[3] [3]

Bourgain, Pointwise ergodic theorems for arithmetic sets , Publ

J. Bourgain, Pointwise ergodic theorems for arithmetic sets , Publ. Math. IHES. 69 (1989), 5-41

work page 1989

[4] [4]

Bourgain, M

J. Bourgain, M. Mirek, E. M. Stein, B. Wr´ obel, On dimension-free variational inequalities for averaging operators in Rd, Geom. Funct. Anal. 28 (2018), no. 1, 58-99

work page 2018

[5] [5]

Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$

J. Bourgain, M. Mirek, E. M. Stein, B. Wr´ obel, Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in Zd, arXiv:1804.07679 (to appear)

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

D. L. Burkholder, A geometric condition that implies the existence of certain singular in- tegrals of Banach-space-valued functions , Conference on harmonic analysis in honor of An- toni Zygmund, Vol. I, II(Chicago, III., 1981), 270-286, Wadswo rth Math. Ser., Wadsworth, Belmont, CA, 1983

work page 1981

[7] [7]

J. T. Campbell, R. L. Jones, K. Reinhold, M. Wierdl, Oscillation and variation for the Hilbert transform, Duke Math. J. 105 (2000), no. 1, 59-83

work page 2000

[8] [8]

J. T. Campbell, R. L. Jones, K. Reinhold, M. Wierdl, Oscillation and variation for singular integrals in higher dimensions , Trans. Amer. Math. Soc. 355 (2003), no. 5, 2115-2137

work page 2003

[9] [9]

Chen, Y, Ding, G

Y. Chen, Y, Ding, G. Hong, H. Liu, Weighted jump and variational inequalities for rough operators, J. Funct. Anal. 274 (2018), no. 8, 2446-2475

work page 2018

[10] [10]

Y, Ding, G. Hong, H. Liu, Jump and variational inequalities for rough operators , J. Fourier Anal. Appl. 23 (2017), no. 3, 679-711

work page 2017

[11] [11]

Y. Do, C. Muscalu, C. Thiele, Variational estimates for paraproducts , Rev. Mat. Iberoam. 28 (2012), no. 3, 8578-78

work page 2012

[12] [12]

A. T. Gillespie, J. L. Torrea, Dimension free estimates for the oscillation of Riesz trans - forms, Isreal J. Math. 141 (2004), 125-144

work page 2004

[13] [13]

Grafakos, Classical and Modern Fourier Analysis , Pearson/Prentice Hall, Upper-Saddle River, 2004

L. Grafakos, Classical and Modern Fourier Analysis , Pearson/Prentice Hall, Upper-Saddle River, 2004

work page 2004

[14] [14]

G. Hong, T. Ma, Vector valued q-variation for ergodic averages and analytic semigroups , J. Math. Anal. Appl. 437 (2016), no. 2, 1084-1100

work page 2016

[15] [15]

G. Hong, T. Ma, Vector valued q-variation for diﬀerential operators and semigroups I , Math. Z. 286 (2017), no. 1-2, 89-120. VECTOR-V ALUED q-V ARIATIONAL INEQUALITIES... 13

work page 2017

[16] [16]

T. P. Hyt¨ onen, Littlewood-Paley-Stein theory for semigroups in UMD space s, Rev. Mat. Iberoam. 23 (2007), no. 3, 973-1009

work page 2007

[17] [17]

T. P. Hyt¨ onen, M. T. Lacey, C. P´ erez,Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), no. 3, 529-540

work page 2013

[18] [18]

T. P. Hyt¨ onen, M. T. Lacey, I. Parissis, A variation norm Carleson theorem for vector- valued Walsh-Fourier series , Rev. Mat. Iberoam. 30 (2014), no. 3, 979-1014

work page 2014

[19] [19]

T. P. Hyt¨ onen, V. N. Jan, M. Veraar, L. Weis, Analysis in Banach spaces. Vol. I. Mar- tingales and Littlewood-Paley theory , Springer, Cham, 2016

work page 2016

[20] [20]

R. L. Jones, R. Kaufman, J. M. Rosenblatt, M. Wierdl, Oscillation in ergodic theory . Ergodic Theory Dynam. Systems 18 (1998), no. 4, 889-935

work page 1998

[21] [21]

R. L. Jones, J. M. Rosenblatt, M. Wierdl, Oscillation inequalities for rectangles , Proc. Am. Math. Soc. 129 1349-1358 (2000)

work page 2000

[22] [22]

R. L. Jones, J. M. Rosenblatt, M. Wierdl, Oscillation in ergodic theory: higher dimensional results, Israel J. Math. 135 (2003), 1-27

work page 2003

[23] [23]

R. L. Jones, R. Kaufman, J. M. Rosenblatt, M. Wierdl, Oscillation and variation for singular integrals in higher dimensions , Trans. Amer. Math. Soc. 355 (2003), no. 5, 2115- 2137.RG

work page 2003

[24] [24]

R. L. Jones, A. Seeger, J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6711-6742

work page 2008

[25] [25]

R. L. Jones, G. Wang, Variation inequalities for the Fej´ er and Poisson kernels , Trans. Amer. Math. Soc. 356 (2004), no. 11, 4493-4518

work page 2004

[26] [26]

Le Merdy, Q

C. Le Merdy, Q. Xu, Strong q -variation inequalities for analytic semigroups , Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2069-2097 (2013)

work page 2012

[27] [27]

Krause, P

B. Krause, P. Zorin-Kranich, Weighted and vector-valued variational estimates for ergo dic averages, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 244-256

work page 2018

[28] [28]

L´ epingle, La variation d′ordre p des semi-martingales , Z

D. L´ epingle, La variation d′ordre p des semi-martingales , Z. Wahrsch. Verw. Gebiete 36 (1976), no. 4, 295-316

work page 1976

[29] [29]

T. Ma, J. L. Torrea, Q. Xu, Weighted variation inequalities for diﬀerential operator s and singular integrals, J. Funct. Anal. 268 (2015), no. 2, 376-416

work page 2015

[30] [30]

T. Ma, J. L. Torrea, Q. Xu, Weighted variation inequalities for diﬀerential operator s and singular integrals in higher dimensions , Sci. China Math. 60 (2017), no. 8, 1419-1442

work page 2017

[31] [31]

Mart ´ ınez, J

T. Mart ´ ınez, J. L. Torrea, Q, Xu, Vector-valued Littlewood-Paley-Stein theory for semi- groups, Adv. Math. 203 (2006), no. 2, 430-475

work page 2006

[32] [32]

Mirek, E

M. Mirek, E. Stein, B. Trojan, ℓp(Zd)-estimates for discrete operators of Radon type: variational estimates , Invent. Math. 209 (2017), no. 3, 665-748

work page 2017

[33] [33]

Mirek, B

M. Mirek, B. Trojan, P. Zorin-Kranich, Variational estimates for averages and truncated singular integrals along the prime numbers , Trans. Amer. Math. Soc. 369 (2017), no. 8, 5403-5423

work page 2017

[34] [34]

Oberlin, A

R. Oberlin, A. Seeger, T. Tao, C. Thiele, J. Wright, A variation norm Carleson theorem , J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 421-464

work page 2012

[35] [35]

Pisier, Martingale with values in uniformly convex spaces , Israel J

G. Pisier, Martingale with values in uniformly convex spaces , Israel J. Math. 20(1975), no. 3-4, 326-350

work page 1975

[36] [36]

Pisier, Q

G. Pisier, Q. Xu, The strong p-variation of martingale and orthogonal series , Probab. Theory Related ﬁelds 77 (1988), no. 4, 497-514

work page 1988

[37] [37]

Xu, Littlewood-Paley theory for functions with values in unifo rmly convex spaces , J

Q. Xu, Littlewood-Paley theory for functions with values in unifo rmly convex spaces , J. Reine Angew. Math. 504 (1998), 195-226. 14 GUIXIANG HONG, WEI LIU, AND TAO MA School of Mathematics and Statistics, Wuhan University, Wu han 430072 and Hubei Key Laboratory of Computational Science, Wuhan Unive rsity, Wuhan 430072, China E-mail address : guixiang.hon...

work page 1998