Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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HARDY TYPE ESTIMATES FOR RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS ON THE HEISENBERG GROUP

  • Gao, Chunfang (School of Mathematics and Statistics Qingdao University)
  • Received : 2020.08.29
  • Accepted : 2021.01.15
  • Published : 2022.03.01
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Abstract

Let ℍn be the Heisenberg group and Q = 2n + 2 be its homogeneous dimension. Let 𝓛 = -∆ℍn + V be the Schrödinger operator on ℍn, where ∆ℍn is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q_1}$ for q1 ≥ Q/2. Let Hp𝓛(ℍn) be the Hardy space associated with the Schrödinger operator 𝓛 for Q/(Q+𝛿0) < p ≤ 1, where 𝛿0 = min{1, 2 - Q/q1}. In this paper, we consider the Hardy type estimates for the operator T𝛼 = V𝛼(-∆ℍn + V )-𝛼, and the commutator [b, T𝛼], where 0 < 𝛼 < Q/2. We prove that T𝛼 is bounded from Hp𝓛(ℍn) into Lp(ℍn). Suppose that b ∈ BMO𝜃𝓛(ℍn), which is larger than BMO(ℍn). We show that the commutator [b, T𝛼] is bounded from H1𝓛(ℍn) into weak L1(ℍn).

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Acknowledgement

This work was financially supported by Shandong Natural Science Foundation of Chian ZR2017JL008.

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