# 36 KAM Mathematical Colloquium

## SOBOLEV SPACES AS BANACH SPACES

May 12, 2000 Lecture Room S6, Charles University, Malostranske nam. 25, Praha 1 10:30 AM

## Abstract

A survey of Banach space properties of most simple classical Sobolev Spaces in $L^p$-norms ($1\le p\le\infty$) defined on (open) subsets of  $R^n$ and compact manifolds, especially on tori, is given. While for $1<p<\infty$ the Sobolev spaces in question are isomorphic to corresponding classical spaces $L^p$, the situation is different in the limit cases $p=1$ and $p=\infty$, for $k$-times continuously differentiable functions and the Sobolev measures in two or more variables. Pathological properties of these spaces related to the failure of the  Grothendieck Theorem on absolutely summing operators are discussed. The proofs involve various analytic tools like Sobolev Embedding Type theorems, Theory of Fourier Multipliers, Whitney and Jones simultaneous extension theorems. Various results due to Grothendieck, Henkin, Mtyagin, Kislyakov, Sidorenko, Bourgain,  Berkson, M. Wojciechowski and the author are discussed.