Public defence, Mathematics and Statistics, MSc Aapo Laukkarinen

Title of the thesis: Convex body domination and its applications to matrix weights and commutators

Thesis defender: Aapo Laukkarinen 
Opponent: Professor Andreas Seeger, University of Wisconsin-Madison, United States
Custos: Academy Professor Tuomas Hytönen, Aalto University School of Science

How can we determine how strongly a mathematical operator changes a function when the result is also influenced by vectors pointing in different directions? This question is central to this doctoral research, which extends the application of a recently introduced method to new settings. 

Mathematical operators describe rules for transforming functions into new functions. They are studied, for example, in harmonic analysis, whose methods are also needed to address many problems in other areas of mathematics. This dissertation investigates mathematical operators in weighted spaces, where the weighting can also depend on the directions of vectors. Analyzing such situations requires methods that can account for this directional variation. 

The research develops a method that helps address this problem by taking vector directions better into account. The method also allows researchers to study a wider range of mathematical operators. The research establishes new results on the conditions under which these operators behave in a controlled manner.

The dissertation also examines what happens when the order of two mathematical operations is reversed. Studying these situations helps us understand the relationships between the operations. The dissertation establishes new results on inequalities associated with such structures and shows when these structures can be approximated effectively. To the best of our knowledge, one of these results is the first of its kind in the study of singular integral operators acting on matrix-weighted spaces. 

This research expands the mathematical tools available for analyzing the behavior of complex operators. The results are part of fundamental research, and their significance lies primarily in helping mathematicians gain a better understanding of operator properties and develop new methods for studying them.

Keywords: Convex body domination, Matrix weight, Commutator

Thesis available for public display 7 days prior to the defence at Aalto University's public display page.