Public defence in Mathematics, M.Sc. (Tech) / M.Soc.Sc. Emma-Karoliina Kurki

Bringing together neglected aspects of weight functions

Title of the doctoral thesis: Weight theory on bounded domains and metric measure spaces

Opponent: Professor Sheldy Ombrosi, Universidad Nacional der Sur, Argentina
Custos: Professori Juha Kinnunen, Aalto University School of Science, Department of Mathematics and Systems Analysis

The public defence will be organised on campus.

The doctoral thesis is publicly displayed 10 days before the defence in the publication archive Aaltodoc of Aalto University.

Electronic thesis

Public defence announcement:

A weight function describes an unequal distribution of mass. Muckenhoupt weights are a class of weight functions that are well-behaved in the sense that their oscillation is limited. Muckenhoupt weights are an indispensable part of the toolkit of modern harmonic analysis, and have important applications to neighboring fields of mathematics. One example is studying the regularity of solutions to partial differential equations, which in turn are the quasi-universal language of physics and mathematical modelling.

The present thesis develops the theory of locally defined weight functions on bounded domains of the Euclidean space, as well as weights on more general metric spaces where the facts of classical geometry are not necessarily true. A cohesive treatment of these cases has been lacking since the introduction of Muckenhoupt weights 50 years ago. One benefit of working in abstract metric spaces is that the structure of the problem is laid bare, ideally allowing us to determine the minimal conditions for a statement to hold true. Furthermore, the methods developed are useful in mathematical analysis in nonlinear environments such as groups and graphs.

On certain domains of the Euclidean space, we show a Poincaré inequality involving a different weight on each side. Poincaré inequalities are essential in the regularity theory of partial differential equations. The proof applies dyadic techniques that have lately been influential in the field of harmonic analysis. On metric measure spaces, we show that a Muckenhoupt weight defined on a measurable subset can be extended into the whole space under certain conditions. Furthermore, we investigate other possible ways to characterize Muckenhoupt-type weights.

Contact details of the doctoral student: [email protected]

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