## Banach Journal of Mathematical Analysis

### Boundedness of Hausdorff operators on Hardy spaces in the Heisenberg group

#### Abstract

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

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

Dates
Accepted: 5 March 2018
First available in Project Euclid: 22 June 2018

https://projecteuclid.org/euclid.bjma/1529632824

Digital Object Identifier
doi:10.1215/17358787-2018-0006

Mathematical Reviews number (MathSciNet)
MR3858754

Zentralblatt MATH identifier
06946296

#### Citation

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. https://projecteuclid.org/euclid.bjma/1529632824

#### References

• [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.