Let be a Schrödinger operator on the Heisenberg group , where is the sub-Laplacian on and the nonnegative potential belongs to the reverse Hölder class with . Here, is the homogeneous dimension of . Assume that is the heat semigroup generated by . The semigroup maximal function related to the Schrödinger operator is defined by . The Riesz transform associated with the operator is defined by , and the dual Riesz transform is defined by , where is the gradient operator on . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator on . Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators , , and acting on the Morrey spaces. In addition, it is shown that the Riesz transform is of weak-type . It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.

中文翻译：

---

海森堡群上与薛定er算子相关的半群极大函数，Riesz变换和Morrey空间

让 成为海森堡集团的Schrödinger算子，其中次拉普拉斯算子在其中，非负势属于反向Hölder类，其。这里，是的齐次维。假设是由产生的热半群。与Schrödinger算子有关的半群极大函数定义为。与运算符关联的Riesz变换定义为，并且双重Riesz变换定义为，上的梯度算子在哪里。在本文中，作者首先介绍了与Schrödinger算子有关的一类Morrey空间。然后，通过使用相关的非负电位内核的一些逐点估计，笔者建立了这些运营商的有界性，，并作用于Morrey空间。此外，它表明Riesz变换 是弱型 。可以证明，对于广义Morrey空间上的这些算子，同样的结论也是正确的。