Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups

Zhiyong Wang; Kai Zhao; Pengtao Li; Yu Liu; Zhiyong Wang; Kai Zhao; Pengtao Li; Yu Liu

doi:10.3934/cam.2023020

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 410-435. doi: 10.3934/cam.2023020

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Research article

Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups

1.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China
2.
School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

Received: 14 June 2023 Revised: 05 July 2023 Accepted: 06 July 2023 Published: 13 July 2023
22E30, 42B35

In this paper, we consider a Schrödinger operator $ L = -\Delta_{\mathbb{H}}+V $ on the stratified Lie group $ \mathbb{H} $. First, we establish fractional heat kernel estimates related to $ L^{\beta} $, $ \beta\in(0, 1) $. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $ BMO_{L}^{v}(\mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {\bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(\mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{\gamma, \theta}_{p, \kappa}(\mathbb{H}) $ and the weak Morrey space $ WL^{\gamma, \theta}_{1, \kappa}(\mathbb{H}) $, respectively.
- Schrödinger operators,
- Morrey spaces,
- Carleson measures,
- fractional heat semigroups,
- Campanato spaces
Citation: Zhiyong Wang, Kai Zhao, Pengtao Li, Yu Liu. Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups[J]. Communications in Analysis and Mechanics, 2023, 15(3): 410-435. doi: 10.3934/cam.2023020

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Abstract

In this paper, we consider a Schrödinger operator $ L = -\Delta_{\mathbb{H}}+V $ on the stratified Lie group $ \mathbb{H} $. First, we establish fractional heat kernel estimates related to $ L^{\beta} $, $ \beta\in(0, 1) $. By utilizing kernel estimations and the fractional Carleson measure, we are able to derive a characterization of the Campanato type space $ BMO_{L}^{v}(\mathbb{H}) $. Second, we demonstrate that both Littlewood-Paley $ {\bf g} $-functions and area functions are bounded on $ BMO^{v}_{L}(\mathbb{H}) $. Finally, we also obtain that the above square functions are bounded on the Morrey space $ L^{\gamma, \theta}_{p, \kappa}(\mathbb{H}) $ and the weak Morrey space $ WL^{\gamma, \theta}_{1, \kappa}(\mathbb{H}) $, respectively.

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