Littlewood-Paley Theory for Triangle Buildings

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Operator Theory

Background:

• The study is situated within the field of harmonic analysis, specifically focusing on function spaces and operator theory.
• It builds upon existing research concerning maximal and square functions in the context of non-commutative harmonic analysis and operator-valued function spaces.

Purpose of the Study:

• To define and investigate the boundedness properties of maximal and square functions associated with a natural two-parameter filtration on the boundary of a triangle building.
• To extend the analysis to the boundedness of martingale transforms in this setting.
• To explore the connection between specific types of buildings and the p-adic Heisenberg group.

Main Methods:

• Definition of maximal and square functions tailored to the two-parameter filtration on the triangle building's boundary.
• Application of techniques from harmonic analysis and functional analysis to establish boundedness results on L^p spaces.
• Analysis of martingale transforms using methods relevant to operator theory and non-commutative settings.

Main Results:

• The boundedness of the defined maximal and square functions on L^p spaces for 1 < p < infinity.
• Demonstration of the p-adic Heisenberg group identification for buildings of a specific type (p-adic).
• Insights into the behavior of martingale transforms within the studied framework.

Conclusions:

• The research successfully establishes key boundedness properties for novel maximal and square functions.
• It provides a significant link between geometric structures (triangle buildings) and abstract algebraic groups (p-adic Heisenberg group).
• The findings contribute to a deeper understanding of martingale transforms and function spaces in advanced harmonic analysis.