#
36 KAM Mathematical Colloquium

##
Prof. ALEKSANDER PELCZYNSKI

###
Polish Academy of Sciences, Warsaw

##
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.