Banach Journal of Mathematical Analysis

Weak boundedness of operator-valued Bochner–Riesz means for the Dunkl transform

Maofa Wang, Bang Xu, and Jian Hu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider operator-valued Bochner–Riesz means with weight function hκ2 under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means BRδ(hκ2;f)(x) for δ>λκ:=d12+j=1dκj. Furthermore, we also obtain the corresponding pointwise convergence theorem.

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 1064-1083.

Received: 25 January 2018
Accepted: 12 April 2018
First available in Project Euclid: 11 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L52: Noncommutative function spaces 32C05: Real-analytic manifolds, real-analytic spaces [See also 14Pxx, 58A07]

Dunkl transform von Neumann algebra Bochner–Riesz means


Wang, Maofa; Xu, Bang; Hu, Jian. Weak boundedness of operator-valued Bochner–Riesz means for the Dunkl transform. Banach J. Math. Anal. 12 (2018), no. 4, 1064--1083. doi:10.1215/17358787-2018-0012.

Export citation


  • [1] Y. Chen, Y. Ding, G. Hong, and H. Liu, Weighted jump and variational inequalities for rough operators, J. Funct. Anal. 274 (2018), no. 8, 2446–2475.
  • [2] Z. Chen, Q. Xu, and Z. Yin, Harmonic analysis on quantum tori, Comm. Math. Phys. 322 (2013), no. 3, 755–805.
  • [3] M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162.
  • [4] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311, no. 1 (1989), 167–183.
  • [5] F. Hansen, An operator inequality, Math. Ann. 246 (1979/1980), no. 3, 249–250.
  • [6] G. Hong, The behavior of the bounds of matrix-valued maximal inequality in ${\mathbb{R}}^{n}$ for large $n$, Illinois. J. Math. 57 (2013), no. 3, 855–869.
  • [7] G. Hong, The behaviour of square functions from ergodic theory in $L^{\infty}$, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4797–4802.
  • [8] G. Hong, M. Junge, and J. Parcet, Algebraic Davis decomposition and asymmetric Doob inequalities, Comm. Math. Phys. 346 (2016), no. 3, 995–1019.
  • [9] G. Hong, M. Junge, and J. Parcet, Asymmetric Doob inequalities in continuous time, J. Funct. Anal. 273 (2017), no. 4, 1479–1503.
  • [10] G. Hong and T. Ma, Vector valued $q$-variation for ergodic averages and analytic semigroups, J. Math. Anal. Appl. 437 (2016), no. 2, 1084–1100.
  • [11] G. Hong and T. Ma, Vector valued $q$-variation for differential operators and semigroups, I, Math. Z. 286 (2017), no. 1–2, 89–120.
  • [12] Y. Jiao and M. Wang, Noncommutative harmonic analysis on semigroups, Indiana Univ. Math. J. 66 (2017), no. 2, 401–417.
  • [13] M. Junge, Doob’s inequality for non-commutative martingales, J.Reine Angew. Math. 549 (2002), 149–190.
  • [14] M. Junge and Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), no. 2, 385–439.
  • [15] T. Ma, J. L. Torrea, and Q. Xu, Weighted variation inequalities for differential operators and singular integrals, J. Funct. Anal. 268 (2015), no. 2, 376–416.
  • [16] T. Ma, J. L. Torrea, and Q. Xu, Weighted variation inequalities for differential operators and singular integrals in higher dimensions, Sci. China Math. 60 (2017), no. 8, 1419–1442.
  • [17] T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188 (2007), no. 881.
  • [18] G. Pisier, Espaces $L_{p}$ non commutatifs à valeurs vectorielles et applications complètement $p$-sommantes, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 10, 1055–1060.
  • [19] G. Pisier and Q. Xu, “Noncommutative $L^{p}$ spaces” in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459–1517.
  • [20] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), no. 3, 445–463.
  • [21] M. Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355, no. 6 (2003), 2413–2438.
  • [22] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Math. Ser. 32, Princeton Univ. Press, Princeton, 1971.
  • [23] S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55.
  • [24] S. Thangavelu and Y. Xu, Riesz transform and Riesz potentials for Dunkl transform, J. Comput. Appl. Math. 199 (2007), no. 1, 181–195.
  • [25] Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), no. 1, 175–192.