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

Abstract

Let $\mathcal{L}=-{\Delta }_{{ℍ}^n}+V$ be a Schrödinger operator on the Heisenberg group ${ℍ}^n$ , where ${\Delta }_{{ℍ}^n}$ is the sub-Laplacian on ${ℍ}^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class ${\mathcal{B}}_q$ with $q\in \left[Q/2,\infty \right)$ . Here, $Q=2n+2$ is the homogeneous dimension of ${ℍ}^n$ . Assume that ${\left\{e^{-t\mathcal{L}}\right\}}_{t>0}$ is the heat semigroup generated by $\mathcal{L}$ . The semigroup maximal function related to the Schrödinger operator $\mathcal{L}$ is defined by ${\mathcal{T}}_{\mathcal{L}}^{\ast }\left(f\right)\left(u\right)≔{\sup }_{t>0}\left|e^{-t\mathcal{L}}f\left(u\right)\right|$ . The Riesz transform associated with the operator $\mathcal{L}$ is defined by ${\mathcal{R}}_{\mathcal{L}}={\nabla }_{{ℍ}^n}{\mathcal{L}}^{-1/2}$ , and the dual Riesz transform is defined by ${\mathcal{R}}_{\mathcal{L}}^{\ast }={\mathcal{L}}^{-1/2}{\nabla }_{{ℍ}^n}$ , where ${\nabla }_{{ℍ}^n}$ is the gradient operator on ${ℍ}^n$ . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator $\mathcal{L}$ 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 ${\mathcal{T}}_{\mathcal{L}}^{\ast }$ , ${\mathcal{R}}_{\mathcal{L}}$ , and ${\mathcal{R}}_{\mathcal{L}}^{\ast }$ acting on the Morrey spaces. In addition, it is shown that the Riesz transform ${\mathcal{R}}_{\mathcal{L}}={\nabla }_{{ℍ}^n}{\mathcal{L}}^{-1/2}$ is of weak-type $\left(1,1\right)$ . It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.