Commutators associated with Schrödinger operators on the nilpotent Lie group

Area of Science:

• Harmonic analysis
• Lie group theory
• Partial differential equations

Background:

• The study focuses on Schrödinger operators on nilpotent Lie groups, utilizing the sub-Laplacian and potentials from the reverse Hölder class.
• The dimension at infinity of the Lie group is a key parameter in the analysis.

Purpose of the Study:

• To derive estimates for the commutator of the Riesz transform associated with the Schrödinger operator.
• To investigate these estimates within a function space larger than the classical Lipschitz space.

Main Methods:

• Analysis of the Schrödinger operator [Formula: see text] on nilpotent Lie groups.
• Application of techniques related to the Riesz transform [Formula: see text].
• Characterization of function spaces beyond Lipschitz spaces.

Main Results:

• The paper obtains estimates for the commutator [Formula: see text].
• These results are established for a function space that is a generalization of the Lipschitz space.
• The findings contribute to the understanding of operators on Lie groups with potentials.

Conclusions:

• The derived estimates provide new insights into the behavior of commutators involving Riesz transforms on nilpotent Lie groups.
• The work extends existing results to a broader class of functions, enhancing the applicability of harmonic analysis techniques.