RG Analysis and Partial Differential Equations

V5B7: Advanced Topics in Analysis - Sobolev Spaces

Winter Semester 2018/2019

Dr. Olli Saari


  • Mo 12-14, 0.011
  • Wed 12-14, 0.011
Notes on the lectures


A preliminary selection
  1. Review of real analysis,
    1. Lp spaces and duality
    2. Distributions
    3. Definition of Sobolev spaces
  2. Generalized Poincaré inequalities
    1. Poincaré's inequality
    2. Self-improving and local Sobolev embeddings
    3. Maximal functions measuring smoothness
    4. Hardy-Sobolev spaces
    5. Pointwise characterizations
  3. Interpolation
    1. Fractional order of smoothness and potentials
    2. Real and complex interpolation
    3. Besov and Triebel-Lizorkin scales
    4. Embeddings
  4. Fine properties
    1. Hausdorff measure
    2. Modulus and Capacity
    3. Precise representatives and absolute continuity
  5. Traces and extensions
  6. Some applications


Lebesgue measure and integration, functional analysis (Banach spaces and operators), basic knowledge about the Fourier transform


The first exams take place on 28.2. and the other exam period is 27.3.-29.3.2019.


  • R.A. Adams, Sobolev spaces, 1975.
  • Adams and Hedberg, Function Spaces and Potential Theory, 1999.
  • Bergh and Löfström, Interpolation Spaces. An Introduction, 1976.
  • A. and J. Björn, Nonlinear Potential Theory on Metric Spaces, 2011
  • Evans and Gariepy, Measure Theory and Fine Properties of Functions, 1991.
  • G. Leoni, A First Course in Sobolev spaces, 2009.