Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups. (21st November 2020) - Wang, Hua
Abstract : Let L = − Δ ℍ n + V be a Schrödinger operator on the Heisenberg group ℍ n, where Δ ℍ n is the sub-Laplacian on ℍ n and the nonnegative potential V belongs to the reverse Hölder class B q with q ∈ Q / 2, ∞ . Here, Q = 2 n + 2 is the homogeneous dimension of ℍ n . Assume that e − t L t > 0 is the heat semigroup generated by L . The semigroup maximal function related to the Schrödinger operator L is defined by T L ∗ f u ≔ sup t > 0 e − t L f u . The Riesz transform associated with the operator L is defined by R L = ∇ ℍ n L − 1 / 2, and the dual Riesz transform is defined by R L ∗ = L − 1 / 2 ∇ ℍ n, where ∇ ℍ n is the gradient operator on ℍ n . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍ n . Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators T L ∗, R L, and R L ∗ acting on the Morrey spaces. In addition, it is shown that the Riesz transform R L = ∇ ℍ n L − 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.