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http://dx.doi.org/10.4134/JKMS.j200484

HARDY TYPE ESTIMATES FOR RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS ON THE HEISENBERG GROUP  

Gao, Chunfang (School of Mathematics and Statistics Qingdao University)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.2, 2022 , pp. 235-254 More about this Journal
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).
Keywords
Heisenberg group; Schrodinger operator; Riesz transform; commutator;
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