Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups - Hua Wang (Journal of Function Spaces)
Let L=−Δℍn+V be a Schrödinger operator on the Heisenberg group ℍn, where Δℍn is the sub-Laplacian on ℍn and the nonnegative potential V belongs to the reverse Hölder class Bq with q∈Q/2,∞. Here, Q=2n+2 is the homogeneous dimension of ℍn. Assume that e−tLt>0 is the heat semigroup generated by L. The semigroup maximal function related to the Schrödinger operator L is defined by TL∗fu≔supt>0e−tLfu. The Riesz transform associated with the operator L is defined by RL=∇ℍnL−1/2, and the dual Riesz transform is defined by RL∗=L−1/2∇ℍn, where ∇ℍn is the gradient operator on ℍn. In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍn. Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators TL∗, RL, and RL∗ acting on the Morrey spaces. In addition, it is shown that the Riesz transform RL=∇ℍnL−1/2 is of weak-type 1,1. It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.