Oleg V. Besov
 Steklov Mathematical Institute, Moscow / Russia

Spaces of differentiable functions
Famous books by Professor Hans Triebel (FSU, Jena), to
whom the Conference is dedicated contain an excellent exposition of and
detailed historical remarks on the theory of function spaces. This theory
was essentially developed by mathematicians of Russia (= fSU): S.L. Sobolev
(spaces $W^k_p$ ), S.M. Nikol'skiy (classes $H^r_p$), P.I. Lizorkin (Liouville
spaces $L^r_p$), O.V. Besov, V.P, Il'in, L.D. Kudryavtzev, G.A. Kalyabin
and many others. The results of these investigations are
summarized in their books.
The aim of this talk is to give a survey of new achievements related to
various aspects of function spaces: imbeddings and equivalent normings,
description in terms of smoothness properties; decompositions and
approximations; interpolation via real and complex methods; trace problems;
extension operators for regular and irregular domains; capacity estimates,
etc.
We shall be focused mainly on the two most important and useful scales of
function spaces  $B^r_{pq}$ and $F^r_{pq}, 1 \le p,q \le \infty, r>0 $
 intensively studied by H. Triebel. The spaces of
generalized smoothness will be also under consideration.
This is joint work with Gennadiy A. Kalyabin.

Gérard Bourdaud
 Université Paris 6, Paris / France

Functional Calculus on Zygmund and BMO spaces
We give a full characterization of
functions $f$ such that the superposition operator $g\mapsto f\circ g$ acts
on the Zygmund spaces $B^m_{\infty\infty}$, for $m$ positive integer, and on
$BMO({\bf R}^n$. The continuity and differentiability of
such operator is also discussed.

Djairo G. De Figueiredo
 Universidade Estadual de
Campinas, Campinas / Brazil

On an inequality of TrudingerMoser and related elliptic equations
It has been shown by Trudinger and Moser that for normalized functions $u$
of the Sobolev space $W^{1,N}(\Omega)$, where $\Omega$ is a domain in
$\R^{N}$, the integral $\int_{\Omega}exp(u^{\alpha_{N}\frac{N}{N1}})dx$
remains bounded. Here $\alpha_{N}=N\omega_{N1}^{\frac{1}{N1}}$, where
$\omega_{N1}$ is the $(N1)$dimensional surface of the unit sphere in
$\R^{N}$. Carleson and Chang proved that there exists a corresponding
extremal function in the case that $\Omega$ is the unit ball in $\R^{N}$. In
this joint work with J.M. do O' and B. Ruf we give a new proof, a
generalization, and a new interpretation of this result. In particular, we
give an explicit sequence which is maximizing for the above integral among
all normalized "concentrating sequences".

Ron De Vore
 University of South Carolina, Columbia / U.S.A.

Some new ideas in interpolation of
operators
A common method for finding interpolation spaces by the real
method of interpolation is to
use retracts. We introduce new ideas on how to use retracts in conjunction with
wavelet decompositions
in order to establish new results on interpolation spaces. These new results
are then used to prove
new versions of the averaging lemma in kinetic theory and new
GagliardoNirenberg inequalities.
This is joint work with Albert Cohen, Wolfgang Dahmen, and Ingrid Daubechies.

David E. Edmunds
 University of Sussex, Brighton / U.K.

Entropy, embeddings and equations
A survey will be given of some of Hans Triebel's many
contributions to Analysis. Particular attention will be given to entropy
numbers, decompositions in function spaces, embeddings between function
spaces and sharp inequalities, with applications to linear and semilinear
elliptic equations.

Vladimir Mazya
 Linköping University, Linköping / Sweden

The Schrödinger
and the relativistic Schrödinger operators on the energy space :
boundedness and compactness criteria
This is a joint work with I. Verbitsky. We give a complete characterization
of the class of functions (or, more generally, complexvalued distributions)
$Q$ such that the following inequality holds:
$$ \int_{R^n} u(x)^2 Q(x) dx \leq const \int_{R^n} \nabla u(x)^2
dx,$$
where the constant is independent of $u\in C^{\infty}_0(R^n)$. Similar
inequalities are proved for the inhomogeneous Sobolev space $W^1_2(R^n)$.
In other words, we establish a criterion for formboundedness of $Q$ relative
to the Laplacian $\Delta$ under no a priori assumptions on $Q$. For the
Schrödinger operator $L=\Delta +Q$, our criterion describes the class of
admissible perturbations $Q$ such that $L:\ring L^1_2(R^n)\to L _2^{1}(R^n)$.
We also establish similar boundedness and compactness criteria for the
relativistic Schrödinger operator.

Akihiko Miyachi
 Tokyo Woman's Christian University, Tokyo / Japan

Hardy type function spaces and its
application
We first show that the purely real variable method can be applied to
the theory of the weighted Hardy space. We show that the theory can
be developped on arbitrary open subset of the Euclidean space and with
respect to general doubling measures. We next use the same method
to study the Herztype Hardy spaces. The theory of Herztype Hardy
space known so far relied on the basic $L^{q}$ theory with
$1 < q < \infty$. We show that the theory can be extended to the range
$q \leqq 1$. Finally, as an application, we consider the transplantation
theorem for Jacobi series.

Stanislav I. Pohozaev
 Steklov Mathematical Institute, Moscow / Russia

The general blowup theory for nonlinear
PDE's
We present the new approach to investigation the nonexistence
of global solutions to nonlinear partial differential equations.
This approach is based on the introduction of a nonlinear capacity that is
appropriate to the initial nonlinear problem and, further, on the use of
standard inequalities with parameters for functions in corresponding
function
spaces. The asymptotic analysis of obtained inequalities gives us
unimprovable
conditions of absence of global solutions for the nonlinear problems
considered.
These results were obtained together with Prof. E. Mitidieri (Italy),
A. Tesei (Italy), and L. Veron (France).

Michael Solomyak
 The Weizmann Institute of Science, Rehovot / Israel

Laplacian and Schrödinger operator on regular
metric trees
Metric tree $\Gamma$ is a tree whose edges are regarded as line segments
of finite length. The Laplacian $\Delta$ on $\Gamma$ is
defined as $\Delta u=u''$ on each edge, with the matching
conditions at each vertex which come from the Kirchhoff laws.
We suppose that a vertex $o\in\Gamma$ ({\it{root}} of $\Gamma$)
is singled out, whose degree is $1$. At this point a boundary condition,
say $u(o)=0$, should be specified.
For a point $x\in\Gamma$,
the number $x$ is defined as the distance between $x$ and $o$.
If $x$ is a vertex, then its {\it{generation}} is the number of
edges connecting $x$ and $o$. Generation of an edge is defined as
the generation of its initial point. A tree is called {\it{regular}}
if all the vertices of the same generation are of the same degree and
all the edges of the same generation are of the same length. Evidently,
all the infinite paths starting at any vertex of generation $j$ have
the same length, say $L_j\le\infty$.
It turns out that the Laplacian on a regular tree admits the decomposition
into orthogonal sum of an infinite family of simpler operators. Each
of them can be reduced to a differential operator acting in $\mathcal
L_2(0,L_j)$.
A similar decomposition holds for the Schrödinger operator, provided its
potential $V(x)$ is symmetric, that is depends only on $x$. Sometimes
this decomposition allows one to give the complete description of the
spectrum
of the initial operator.
Let in particular all the vertices $x\not=o$ are of the same degree and
all the edges are of the same length. Then the spectrum of the Laplacian
has the bandgap structure. If the Laplacian is perturbed by a
realvalued decaying potential $\alpha V$ (where $\alpha$ is a large
parameter),
then eigenvalues in the gaps may appear. Study of the behaviour of
the number of these eigenvalues as $\alpha\to\infty$ has much in common
with the study of some spectral problems related to fractals. Several
examples will be presented.

Gunther Uhlmann
 University of Washington, Seattle / U.S.A.

InsideOut: Inverse
Boundary Problems
We survey recent progress in inverse boundary problems. In these problems
one attempts to determine the internal properties of a medium by making
measurements at the boundary of the medium. In particular we will consider
the problem of determining a Riemannian metric on a Riemannian manifold
with boundary by measuring the geodesic distance between boundary points.
We shall consider also the inverse problem of determining a potential or a
conductivity on a bounded domain in Euclidean space from partial
information on the DirichlettoNeumann map for the associated
Schrödinger equation or conductivity equation.
