Interpolation Spaces

  • Lecturer: Jonas Sauer
  • Date: Tuesday, 09:00 - 10:30
  • Room: MPI MiS A3 01
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: Real interpolation, complex interpolation, reiteration theorem, domains of operators
  • Prerequisites: Calculus, functional analysis

Abstract:

Which space lies halfway between \(C([0,1])\) and \(C^1([0,1])\)? Is it the Hölder space \(C^{1/2}([0,1])\)?

Does \(C^1([0,1])\) lie halfway between \(C([0,1])\) and \(C^2([0,1])\)?

These questions lead to the theory of interpolation spaces (and it is fair to say the answers are, somewhat surprisingly, "yes" for the first question but "no" for the second). Also, many trace theorems, domains for fractional powers of operators and maximal regularity spaces for abstract Cauchy problems are obtained from interpolation between suitable Banach spaces.

The plan of this lecture is to

  • show that the \(L^p\)-scale of \(p\)-integrable functions fits in this picture (Theorems of Riesz-Thorin and Marcinkiewicz),
  • introduce the real, trace and complex interpolation methods,
  • establish duality and reiteration theorems,
  • give concrete examples for interpolation spaces (Höolder spaces, Sobolev-Slobodeckii spaces, Besov spaces), and thereby extract elegant proofs for Young's inequality, Sobolev embeddings and trace theorems,
  • investigate interpolation spaces of domains of closed operators and how they enter the study of abstract Cauchy problems.

Access Information

Please use the entry doors Kreuzstr. 7a (rooms A3 01, A3 02) and Kreustr. 7c (rooms G3 10, G2 01), both in the inner court yard, and go to the 3rd. floor (see the map). To reach the Leibniz-Saal use the main entry Inselstr. 22 and go to the 1st. floor.
Please note: The doors will be opened 15 minutes before the lecture starts and closed after beginning of the lecture!


Regular Lectures (Summer 2017)

26.06.2017, 11:43