Czechoslovak Mathematical Journal

, Volume 66, Issue 1, pp 169–191 | Cite as

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

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


Let L 1 = −Δ + V be a Schr:dinger 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 sn/2. The Hardy type space \(H_{{L_2}}^1\) is defined in terms of the maximal function with respect to the semigroup \(\left\{ {{e^{ - t{L_2}}}} \right\}\) 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 = bRf - R(bf) generated by the Riesz transform \(R = {\nabla ^2}L_2^{ - 1/2}\), where \(b \in BM{O_\theta }(\varrho )\), which is larger than the space BMO(ℝ n ). Moreover, we prove that R b is bounded from the Hardy space \(H_{\mathcal{L}_1 }^1 \) into weak \(L_{weak}^1 (\mathbb{R}^n )\).


commutator Hardy space reverse Hölder inequality Riesz transform Schrödinger operator Schrödinger type operator 

MSC 2010

42B35 35J10 42B30 42B20 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  • Yu Liu
    • 1
  • Jing Zhang
    • 1
  • Jie-Lai Sheng
    • 1
  • Li-Juan Wang
    • 1
  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingHaidian, BeijingP.R. China

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