Public defence in the field of Mathematics and Statistics, M.Sc. (Tech) Timo Takala

Title of the thesis: Nonlocal function spaces and conformal deformations of metric measure spaces

Thesis defender: Timo Takala
Opponent:  Associate Professor Matthew Badger, University of Connecticut, US
Custos: Associate Professor Riikka Korte, Aalto University School of Science

The oscillation of functions is an interesting and useful mathematical property. In particular the space of functions of bounded mean oscillation (BMO) is important for example in harmonic analysis and in the study of partial differential equations. The John-Nirenberg space (JNp) is a generalization of BMO, and the oscillation of JNp functions is also bounded in a certain sense. JNp and BMO were introduced in the literature already in the 1960s but only recently there has been more interest in studying JNp. Properties of JNp spaces are studied in this thesis.

JNp and BMO are examples of nonlocal function spaces. Another nonlocal function space studied in this thesis is the Besov space. In boundary value problems of partial differential equations, the solution gets certain values on the boundary of the domain. If the solution is a Sobolev function inside the domain, then the correct space for the boundary values is the corresponding Besov space, which is also called the fractional Sobolev space. Consequently the Besov space is relevant in the study of boundary value problems. Partial differential equations can be studied in Euclidean spaces but also in rather general metric measure spaces, where there is a distance defined between points, but for example the concept of direction does not exist.

Sphericalizations and flattenings are examples of conformal deformations of metric measure spaces. Sphericalizations transform an unbounded space into a bounded space, and flattenings transform a bounded space into an unbounded space. These deformations are useful in the study of partial differential equations, because some methods are applicable in bounded spaces but not in unbounded spaces. Similarly some methods are applicable in unbounded spaces but not in bounded spaces.

In this thesis the preservation of Besov functions in these deformations is studied, so that the deformations can be applied in the study of boundary value problems. The thesis also presents general conditions for sphericalizations that preserve other important geometric properties of metric measure spaces, which enable for example techniques in the study of partial differential equations and in harmonic analysis. Previously different sphericalizations have been constructed for different applications. Our results aim to offer a unified theory for this kind of conformal deformations.

Keywords: John-Nirenberg space, bounded mean oscillation, metric measure space, Besov space, sphericalization, flattening

Contact information: timo.i.takala@aalto.fi 

Thesis available for public display 10 days prior to the defence at Aaltodoc. 

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