Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Abstract EN

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.