Interpolation Spaces

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.

Keywords
Real interpolation, complex interpolation, reiteration theorem, domains of operators

Prerequisites
Calculus, functional analysis

Audience
MSc students, PhD students, Postdocs

Language
English