The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

Geometric Properties of Metric Measure Spaces and Sobolev-Type Inequalities

Riikka Korte

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Information and Natural Sciences for public examination and debate in Auditorium E at Helsinki University of Technology (Espoo, Finland) on the 29th of February, 2008, at 12 noon.

Overview in PDF format (ISBN 978-951-22-9211-0)   [221 KB]
Dissertation is also available in print (ISBN 978-951-22-9210-3)

Abstract

This dissertation studies analysis in metric spaces that are equipped with a doubling measure and satisfy a Poincaré inequality. The treatise consists of four articles in which we discuss geometric implications of the Poincaré inequality as well as equivalent characterizations of capacities, Sobolev inequalities and conditions for the thickness of a boundary of sets. We also study the size of exceptional sets for Newtonian functions.

This thesis consists of an overview and of the following 4 publications:

  1. R. Korte, Geometric implications of the Poincaré inequality, Results in Mathematics, 50 (2007), pp. 93-107.
  2. J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via boxing inequality in metric spaces, Indiana University Mathematics Journal, 57 (2008), to appear.
  3. R. Korte and N. Shanmugalingam, Equivalence and self–improvement of p–fatness and Hardy's inequality, and association with uniform perfectness, arXiv:0709.2013v2 [math.FA]. © 2008 by authors.
  4. J. Kinnunen and R. Korte, Characterizations of Sobolev inequalities on metric spaces, arXiv:0709.1097v2 [math.AP]. © 2008 by authors.

Keywords: boxing inequality, capacity, doubling measure, functions of bounded variation, Hausdorff content, Lebesgue points, metric spaces, modulus, Newtonian spaces, Poincaré inequality, quasiconvexity, Sobolev–Poincaré inequality, Sobolev spaces

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© 2008 Helsinki University of Technology

Last update 2011-05-26