Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Yu Liu; Jing Zhang; Jie-Lai Sheng; Li-Juan Wang

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 169-191
  • ISSN: 0011-4642

Abstract

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Let ℒ 1 = - Δ + V be a Schrödinger operator and let ℒ 2 = ( - Δ ) 2 + V 2 be a Schrödinger type operator on ℝ n ( n ≥ 5 ) , where V ≠ 0 is a nonnegative potential belonging to certain reverse Hölder class B s for s ≥ n / 2 . The Hardy type space H ℒ 2 1 is defined in terms of the maximal function with respect to the semigroup { e - t ℒ 2 } and it is identical to the Hardy space H ℒ 1 1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator ℛ b = b ℛ f - ℛ ( b f ) generated by the Riesz transform ℛ = ∇ 2 ℒ 2 - 1 / 2 , where b ∈ BMO θ ( ρ ) , which is larger than the space BMO ( ℝ n ) . Moreover, we prove that ℛ b is bounded from the Hardy space H ℒ 2 1 ( ℝ n ) into weak L weak 1 ( ℝ n ) .

How to cite

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Liu, Yu, et al. "Some estimates for commutators of Riesz transform associated with Schrödinger type operators." Czechoslovak Mathematical Journal 66.1 (2016): 169-191. <http://eudml.org/doc/276801>.

@article{Liu2016,
abstract = {Let $\mathcal \{L\}_1=-\Delta +V$ be a Schrödinger operator and let $\mathcal \{L\}_2=(-\Delta )^2+V^2$ be a Schrödinger type operator on $\{\mathbb \{R\}^n\}$$(n \ge 5)$, where $V \ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge \{n\}/\{2\}$. The Hardy type space $H^1_\{\mathcal \{L\}_2\}$ is defined in terms of the maximal function with respect to the semigroup $\lbrace \{\rm e\}^\{-t \mathcal \{L\}_2\}\rbrace $ and it is identical to the Hardy space $H^1_\{\mathcal \{L\}_1\}$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator $\mathcal \{R\}_b=b\mathcal \{R\}f-\mathcal \{R\}(bf)$ generated by the Riesz transform $\mathcal \{R\}=\nabla ^2\mathcal \{L\}_2^\{-\{1\}/\{2\}\}$, where $b\in \{\rm BMO\}_\theta (\rho )$, which is larger than the space $\{\rm BMO\}(\mathbb \{R\}^n)$. Moreover, we prove that $\mathcal \{R\}_b$ is bounded from the Hardy space $H_\{\mathcal \{L\}_2\}^1(\mathbb \{R\}^n)$ into weak $L_\{\rm weak\}^1(\mathbb \{R\}^n)$.},
author = {Liu, Yu, Zhang, Jing, Sheng, Jie-Lai, Wang, Li-Juan},
journal = {Czechoslovak Mathematical Journal},
keywords = {commutator; Hardy space; reverse Hölder inequality; Riesz transform; Schrödinger operator; Schrödinger type operator},
language = {eng},
number = {1},
pages = {169-191},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some estimates for commutators of Riesz transform associated with Schrödinger type operators},
url = {http://eudml.org/doc/276801},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Liu, Yu
AU - Zhang, Jing
AU - Sheng, Jie-Lai
AU - Wang, Li-Juan
TI - Some estimates for commutators of Riesz transform associated with Schrödinger type operators
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 169
EP - 191
AB - Let $\mathcal {L}_1=-\Delta +V$ be a Schrödinger operator and let $\mathcal {L}_2=(-\Delta )^2+V^2$ be a Schrödinger type operator on ${\mathbb {R}^n}$$(n \ge 5)$, where $V \ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge {n}/{2}$. The Hardy type space $H^1_{\mathcal {L}_2}$ is defined in terms of the maximal function with respect to the semigroup $\lbrace {\rm e}^{-t \mathcal {L}_2}\rbrace $ and it is identical to the Hardy space $H^1_{\mathcal {L}_1}$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator $\mathcal {R}_b=b\mathcal {R}f-\mathcal {R}(bf)$ generated by the Riesz transform $\mathcal {R}=\nabla ^2\mathcal {L}_2^{-{1}/{2}}$, where $b\in {\rm BMO}_\theta (\rho )$, which is larger than the space ${\rm BMO}(\mathbb {R}^n)$. Moreover, we prove that $\mathcal {R}_b$ is bounded from the Hardy space $H_{\mathcal {L}_2}^1(\mathbb {R}^n)$ into weak $L_{\rm weak}^1(\mathbb {R}^n)$.
LA - eng
KW - commutator; Hardy space; reverse Hölder inequality; Riesz transform; Schrödinger operator; Schrödinger type operator
UR - http://eudml.org/doc/276801
ER -

References

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  1. Bongioanni, B., Harboure, E., Salinas, O., 10.1007/s00041-010-9133-6, J. Fourier Anal. Appl. 17 (2011), 115-134. (2011) Zbl1213.42075MR2765594DOI10.1007/s00041-010-9133-6
  2. Bramanti, M., Cerutti, C. M., Commutators of singular integrals on homogeneous spaces, Boll. Unione Mat. Ital. 10 (1996), 843-883. (1996) MR1430157
  3. Cao, J., Liu, Y., Yang, D., Hardy spaces H ℒ 1 ( ℝ n ) associated to Schrödinger type operators ( - Δ ) 2 + V 2 , Houston J. Math. 36 (2010), 1067-1095. (2010) MR2753734
  4. Coifman, R. R., Rochberg, R., Weiss, G., 10.2307/1970954, Ann. Math. 103 (1976), 611-635. (1976) MR0412721DOI10.2307/1970954
  5. Duong, X. T., Yan, L., 10.1017/S0004972700033669, Bull. Aust. Math. Soc. 67 (2003), 187-200. (2003) Zbl1023.42010MR1972709DOI10.1017/S0004972700033669
  6. Dziubański, J., Zienkiewicz, J., 10.4171/RMI/257, Rev. Mat. Iberoam 15 (1999), 279-296. (1999) MR1715409DOI10.4171/RMI/257
  7. Guo, Z., Li, P., Peng, L., 10.1016/j.jmaa.2007.05.024, J. Math. Anal. Appl. 341 (2008), 421-432. (2008) MR2394095DOI10.1016/j.jmaa.2007.05.024
  8. Janson, S., 10.1007/BF02386000, Ark. Math. 16 (1978), 263-270. (1978) Zbl0404.42013MR0524754DOI10.1007/BF02386000
  9. Li, H. Q., 10.1006/jfan.1998.3347, J. Funct. Anal. 161 (1999), French 152-218. (1999) MR1670222DOI10.1006/jfan.1998.3347
  10. Li, P., Peng, L., 10.1017/S0004972710000390, Bull. Aust. Math. Soc. 82 (2010), 367-389. (2010) Zbl1210.47048MR2737950DOI10.1017/S0004972710000390
  11. Liu, Y., 10.4134/JKMS.2010.47.2.425, J. Korean Math. Soc. 47 (2010), 425-443. (2010) Zbl1187.22008MR2605991DOI10.4134/JKMS.2010.47.2.425
  12. Liu, Y., Dong, J., 10.1007/s11118-009-9143-7, Potential Anal. 32 (2010), 41-55. (2010) Zbl1197.42008MR2575385DOI10.1007/s11118-009-9143-7
  13. Liu, Y., Huang, J. Z., Dong, J. F., 10.1007/s11425-012-4551-3, Sci. China. Math. 56 (2013), 1895-1913. (2013) Zbl1278.42031MR3090862DOI10.1007/s11425-012-4551-3
  14. Liu, Y., Huang, J., Xie, D., 10.1007/s00013-009-0098-0, Arch. Math. 94 (2010), 255-264. (2010) Zbl1189.22006MR2602452DOI10.1007/s00013-009-0098-0
  15. Liu, Y., Sheng, J., 10.1016/j.jmaa.2014.04.053, J. Math. Anal. Appl. 419 (2014), 298-328. (2014) MR3217150DOI10.1016/j.jmaa.2014.04.053
  16. Liu, Y., Wang, L., Dong, J., Commutators of higher order Riesz transform associated with Schrödinger operators, J. Funct. Spaces Appl. 2013 (2013), Article ID 842375, 15 pages. (2013) Zbl1279.47052MR3053277
  17. Shen, Z., 10.5802/aif.1463, Ann. Inst. Fourier (Grenoble) 45 (1995), 513-546. (1995) MR1343560DOI10.5802/aif.1463
  18. Sugano, S., 10.3836/tjm/1184963655, Tokyo J. Math. 30 (2007), 179-197. (2007) MR2328062DOI10.3836/tjm/1184963655
  19. Zhong, J., Harmonic Analysis for some Schrödinger Type Operators, Ph.D. Thesis, Princeton University (1993). (1993) MR2689454

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