# Public defence in Mathematics, M.Sc. (Tech) Kim Myyryläinen

Public defence from the Aalto University School of Science, Department of Mathematics and Systems Analysis

Title of the thesis: Parabolic bounded mean oscillation and Muckenhoupt weights

Doctoral student: Kim Myyryläinen
Opponent: Professor Luboš Pick, Charles University, Prague, Czech Republic
Custos: Professor Juha Kinnunen, Aalto University School of Science, Department of Mathematics and Systems Analysis

Functions of bounded mean oscillation (BMO) and Muckenhoupt weights are essential concepts in modern harmonic analysis. They are used to measure the oscillation of a function at respective scales and to distribute mass to the underlying space. The theory of BMO and Muckenhoupt weights is comprehensive and has many applications also in other fields of mathematics such as partial differential equations.

This thesis studies parabolic bounded mean oscillation and Muckenhoupt weights that are generalizations of the classical time-independent concepts. The objective of the study is to extend the classical theory of BMO and Muckenhoupt weights to the time-dependent parabolic setting. Parabolic BMO and Muckenhoupt weights arise from parabolic partial differential equations, in particular, a doubly nonlinear equation that models nonlinear diffusion. The doubly nonlinear equation is a generalization of the classical heat equation modelling heat conduction.

The results obtained in the thesis include various decay estimates for the oscillation of a function. We show weighted norm inequalities for a parabolic maximal function and obtain a complete theory for the limiting parabolic Muckenhoupt class including characterization and factorization results. The theory for the other endpoint class is also developed and several new characterizations and self-improving phenomena are discovered. The proofs of our results apply covering, chaining and decomposition techniques adapted to the parabolic geometry. In addition to the parabolic framework, other forms of oscillation are considered in the general context of metric measure spaces.

Thesis available for public display 10 days prior to the defence at: https://aaltodoc.aalto.fi/doc_public/eonly/riiputus/

Contact information:

 Email [email protected]

Doctoral theses at the School of Science: https://aaltodoc.aalto.fi/handle/123456789/52

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