Boundedness of operators generated by fractional heat semigroups related to Schrödinger operators on stratified Lie groups via T1 theorem

Abstract

Let \(L=-{\Delta }_{{\mathbb {G}} }+V\) be a Schrödinger operator on the stratified Lie group \({\mathbb {G}},\) where \({\Delta }_{{\mathbb {G}} }\) is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class \(B_{q}, q\ge {\mathcal {Q}}/2,\) in which \({\mathcal {Q}}\) is the homogeneous dimension of \({\mathbb {G}}.\) In this article, we firstly study the fractional heat semigroups \(\{e^{-tL^{\alpha }}\}_{t>0}\) with \(\alpha >0\) associated with L. Subsequently, the regularities of the fractional heat semigroup is estimated with the use of the subordinative formula. Furthermore, in terms of application, we establish the \(BMO_{L}^{\gamma }({\mathbb {G}})\)-boundedness of the maximal function and the Littlewood–Paley \({\mathfrak {g}}\)-functions related with the Schrödinger operator L by T1 theorem, respectively.