Graded hypoellipticity of BGG sequences

Area of Science:

• Differential analysis
• Geometric analysis
• Pseudodifferential calculus

Background:

• Hypoellipticity is a key property in the analysis of partial differential equations.
• Filtered manifolds provide a general framework for studying geometric structures.
• The Rockland criterion and elliptic sequences are fundamental concepts in this field.

Purpose of the Study:

• To extend the Rockland criterion to general filtered manifolds.
• To analyze hypoellipticity and Rockland sequences on these manifolds.
• To apply these findings to the study of Bernstein-Gelfand-Gelfand (BGG) sequences in regular parabolic geometries.

Main Methods:

• Developing a pseudodifferential calculus tailored for filtered manifolds.
• Constructing a parametrix and detailing its analytic structure.
• Generalizing the BGG machinery to non-canonical filtered manifolds.

Main Results:

• The Rockland criterion is successfully extended to general filtered manifolds.
• A precise analytic structure of the parametrix is described.
• Generalized BGG sequences are demonstrated to be Rockland in a graded sense.

Conclusions:

• The study provides a novel framework for analyzing hypoellipticity on filtered manifolds.
• The generalization of BGG sequences offers new insights into parabolic geometries.
• This work bridges concepts from pseudodifferential calculus and geometric analysis.