Banach Journal of Mathematical Analysis

Boundedness of Hausdorff operators on Hardy spaces in the Heisenberg group

Qingyan Wu and Zunwei Fu

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In the setting of the Heisenberg group, we define weighted Hardy spaces by means of their atomic characterization, and we establish the (sharp) boundedness of Hausdorff operators on power-weighted Hardy spaces. Moreover, we obtain sufficient and necessary conditions for the boundedness of Hausdorff operators on local Hardy spaces in the Heisenberg group.

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 909-934.

Received: 23 October 2017
Accepted: 5 March 2018
First available in Project Euclid: 22 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47G10: Integral operators [See also 45P05]
Secondary: 22E25: Nilpotent and solvable Lie groups 26D15: Inequalities for sums, series and integrals 42B30: $H^p$-spaces

Hausdorff operator Heisenberg group Hardy space power weight local Hardy space


Wu, Qingyan; Fu, Zunwei. Boundedness of Hausdorff operators on Hardy spaces in the Heisenberg group. Banach J. Math. Anal. 12 (2018), no. 4, 909--934. doi:10.1215/17358787-2018-0006.

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  • [1] J. Chen, D. Fan, and S. Wang, Hausdorff operators on Euclidean spaces, Appl. Math. J. Chinese Univ. Ser. B 28 (2013), no. 4, 548–564.
  • [2] M. Christ and D. Geller, Singular integral characterizations of Hardy spaces on homogeneous groups, Duke Math. J. 51 (1984), no. 3, 547–598.
  • [3] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. (N.S.) 83 (1977), no. 4, 569–645.
  • [4] T. Coulhon, D. Müller and J. Zienkiewicz, About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (1996), no. 2, 369–379.
  • [5] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, 1989.
  • [6] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton Univ. Press, Princeton, 1982.
  • [7] S. Fridli, Hardy spaces generated by an integrability condition, J. Approx. Theory 113 (2001), no. 1, 91–109.
  • [8] Z. Fu, L. Grafakos, S. Lu, and F. Zhao, Sharp bounds for $m$-linear Hardy and Hilbert operators, Houston J. Math. 38 (2012), no. 1, 225–244.
  • [9] Z. Fu, Z. Liu, and S. Lu, Commutators of weighted Hardy operators in $\mathbb{R}^{n}$, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3319–3328.
  • [10] P. Galanopoulos and A. G. Siskakis, Hausdorff matrices and composition operators, Illinois J. Math. 45 (2001), no. 3, 757–773.
  • [11] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95–153.
  • [12] D. Geller, Some results in $H^{p}$ theory for the Heisenberg group, Duke Math. J. 47 (1980), no. 2, 365–390.
  • [13] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27–42.
  • [14] V. S. Guliev, Two-weighted $L_{p}$-inequalities for singular integral operators on Heisenberg groups, Georgian Math. J. 1 (1994), no. 4, 367–376.
  • [15] J. H. Guo, L. J. Sun, and F. Y. Zhao, Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714.
  • [16] A. Hulanicki, The distribution of energy in the Brownian motion in Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976), no. 2, 165–173.
  • [17] Y. Kanjin, The Hausdorff operators on the real Hardy spaces $H^{p}(\mathbb{R})$, Studia Math. 148 (2001), no. 1, 37–45.
  • [18] A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309–338.
  • [19] R. H. Latter and A. Uchiyama, The atomic decomposition for parabolic $H^{p}$ spaces, Trans. Amer. Math. Soc. 253 (1979), 391–398.
  • [20] A. K. Lerner and E. Liflyand, Multidimensional Hausdorff operators on the real Hardy space, J. Aust. Math. Soc. 83 (2007), no. 1, 79–86.
  • [21] E. Liflyand, Boundedness of multidimensional Hausdorff operators on $H^{1}(\mathbb{R}^{n})$, Acta Sci. Math. (Szeged) 74 (2008), no. 3-4, 845–851.
  • [22] E. Liflyand, Hausdorff operators on Hardy spaces, Eurasian Math. J. 4 (2013), no. 4, 101–141.
  • [23] E. Liflyand and A. Miyachi, Boundedness of the Hausdorff operators in $H^{p}$ spaces, $0<p<1$, Studia Math. 194 (2009), no. 3, 279–292.
  • [24] E. Liflyand and F. Móricz, The Hausdorff operator is bounded on the real Hardy space $H^{1}(\mathbb{R})$, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1391–1396.
  • [25] C. Lin, H. Liu, and Y. Liu, Hardy spaces associated with Schrödinger operators on the Heisenberg group, preprint, arXiv:1106.4960v1 [math.AP].
  • [26] Y. Liu, Compensated compactness and the stratified Lie group, Anal. Theory Appl. 25 (2009), no. 2, 101–108.
  • [27] S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics, World Sci. Publ., Hackensack, N.J., 2007.
  • [28] C. A. Nolder, Hardy-Littlewood inequality for quasiregular maps on Carnot groups, Nonlinear Anal. 63 (2005), no. 5–7, e407–e415.
  • [29] J. Ruan and D. Fan, Hausdorff operators on the power weighted Hardy spaces, J. Math. Anal. Appl. 433 (2016), no. 1, 31–48.
  • [30] J. Ruan, D. Fan, and Q. Wu, Weighted Herz space estimates for Hausdorff operators on the Heisenberg group, Banach J. Math. Anal. 11 (2017), no. 3, 513–535.
  • [31] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • [32] Q. Wu and D. Fan, Hardy space estimates of Hausdorff operators on the Heisenberg group, Nonlinear Anal. 164 (2017), 135–154.
  • [33] Q. Wu and Z. Fu, Weighted $p$-adic Hardy operators and their commutators on $p$-adic central Morrey spaces, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 2, 635–654.
  • [34] X. Wu, Necessary and sufficient conditions for generalized Hausdorff operators and commutators, Ann. Funct. Anal. 6 (2015), no. 3, 60–72.
  • [35] J. Xiao, $L^{p}$ and BMO bounds of weighted Hardy-Littlewood averages, J. Math. Anal. Appl. 262 (2001), no. 2, 660–666.
  • [36] R. Xu and F. Meng, Some new weakly singular integral inequalities and their applications to fractional differential equations, J. Inequal. Appl. 2016, no. 78.