Scientific Programme
Programme
In the morning we will have summer school lectures. After lunch, invited lecture talks will be given, followed by contributed lectures with the following format:
 90 minutes for summer school course lectures;
 45 minutes for invited workshop lectures;
 40 and 30 minutes for contributed talks.
Programme Overview
Summer School Courses
Each course consists of three sessions of one hour and a half, covering the following topics:
 Course A  Sobolev, capacitary and isocapacitary inequalities
 Vladimir Maz'ya  University of Liverpool, UK and University of Linköping, Sweden
abstract:
 Author's results on the equivalence of various isoperimetric and isocapacitary
inequalities, on one hand, and Sobolev's type imbedding and compactness theorems, on
the other hand, will be surveyed. Most of the proofs apply to functions on Riemannian
manifolds and even very general topological spaces, so that these techniques
provide a substitute for the rearrangement methods used to obtain sharp constants in
Sobolev's type inequalities in the Euclidean space.
 Area minimizing function and Sobolev's type inequalities.
 Capacity minimizing function and Sobolev's type inequalities.
 Application of capacities to integral inequalities for functions vanishing on a set.
 Lecture slides,
Lecture material
 Course B  Weighted problems for operators of harmonic analysis in some Banach
function spaces
 Vakhtang Kokilashvili  A. Razmadze Mathematical Institute, Georgia
abstract:
 The aim of our lectures is to discuss the following topics:
 Twoweighted estimates for maximal functions, singular integrals and
potentials in variable exponent Lebesgue spaces.
 Boundedness criteria for the operators of nonlinear harmonic analysis
(Cauchy singular integrals, maximal functions defined on rectifiable
curves) in weighted grand L^{p} spaces.
 Applications to the Fourier Analysis and boundary value problems for
analytic and harmonic functions.

 Lecture notes: Course B
 Course C  Variable Lebesgue Spaces
 David CruzUribe, SFO  Trinity College, Hartford, Connecticut, USA
abstract:

In these lectures we will explore the properties and applications of the variable Lebesgue spaces L^{p(·)}, also known as the variable exponent spaces.
 Banach space properties of the variable Lebesgue spaces. We will consider the structure of these spaces depending on whether the exponent function p(·) is bounded or unbounded, paying particular attention to convergence and duality.
 The HardyLittlewood maximal operator and extrapolation. We will discuss sufficient conditions on the exponent function p(·) for the maximal operator to be bounded and we will show how Rubio de Francia extrapolation can be used to prove that the classical operators of harmonic analysis are bounded on the variable Lebesgue spaces.
 Applications. We will show how the boundedness of the maximal operator and extrapolation can be applied to problems in variable Sobolev spaces and partial differential equations.

 Course C lecture handout: Handout 1, Handout 2, Handout 3
Workshop
Invited Talks
 Bisobolev homeomorphisms and degenerate elliptic systems  Carlo Sbordonne
abstract:
Suppose that f = (u; v) is a homeomorphism between planar domains, such that f and its inverse are of Sobolev class W^{1,1}. We prove that, then, u and v have the same set of critical points a.e.. This allows us to show that to each such f there corresponds a degenerate elliptic matrix A(x) with det A(x) = 1 such that u and v are very weak solutions to the equations div A(x)Du=0 and div A(x)Dv=0. We study planar ACLhomeomorphisms i.e. homeomorphisms f =f(u,v), which are absolutely continuous on lines parallel to the axes together with their inverse. These are results obtained in cooperation with L. Greco, C. Trombetti, N.Fusco, S.Hencl, G. Moscariello, A. Passarelli di Napoli, R. Schiattarella.
 Hardy type inequalities: prehistory, history and some new developments  LarsErik Persson
abstract:
We will first describe something from the dramatic prehistory of 10 years of work until Hardy in 1925 finally proved his famous inequality. In particular, we will shortly describe my own experience when during a summer I wrote the first version of [4] by almost living and feeling as I think Hardy did during this period. After that we will present a "one line convexity proof" of the inequality we now have discovered and developed and which could have changed both the prehistory and history if Hardy had found it. Moreover, we will mention some important problems, results and applications obtained and described in the rich literature in the field, see e.g. the books [1][3] and the references given there. Finally, there will be described some very new ideas and also new results recently obtained in collaboration with Natasha Samko and Stefan Samko.
[1] A. Kufner and L.E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New/JerseyLondon/Singapore/Hong Kong, 2003 (357 pages).
[2] A. Kufner, L. Maligranda and L.E. Persson, The Hardy Inequality. About its History and some Related Results, Vydavatelsky Servis Publishing House, Pilsen, 2007 (161 pages).
[3] A. Meshki, V. Kokalishvili and L.E. Persson, Weighted Norm Inequalities for Integral Transforms with Product Kernels, Nova Scientific Publishers, Inc., Springer, New York, 2009 (329 pages).
[4] A. Kufner, L. Maligranda and L.E. Persson,The prehistory of the Hardy inequality, Amer. Math. Monthly 113 (8), 715732, 2006.
 Nonlinear CalderonZygmund theory for the pLaplacian including variable exponents  Lars Diening
abstract:
It has been possible in the recent years to generalize parts
of the linear CalderonZymund theory to the nonlinear setting of the
pLaplacian. We first discuss the basic principles and results of this
method. After that we show how to generalize this to the context of
variable exponents.
Contributed Talks
last updated: 01062010
 Capacitary Function Spaces and
Interpolation  Pilar Silvestre Albero
 Besov spaces with variable smoothness and integrability  Alexandre Almeida
 Quasilinear elliptic equation with variable exponent coefficients  Sami Aouaoui
 Existence result of sublinear elliptic equation with nonstandard growth conditions  Mohamed Benrhouma
 Real interpolation of generalized BesovHardy spaces and applications  António Caetano
 Some extremal problems of harmonic analysis for positive definite functions  Dmitry Gorbachev
 The Riesz potential as a multilinear operator into BMO_{β} spaces  Silvia Ines Hartzstein
 Singular integral operators on Nakano spaces with weights having finite sets of discontinuities  Alexei Karlovich
 Algebras of pseudodifferential operators with nonregular symbols  Yuri Karlovich
 Representation of a function via the absolutely convergent Fourier integral  Elijah Liflyand
 Hardy type inequalities in variable exponent Lebesque space  Farman Mamedov
 Sharp weighted bounds for Singular Integrals and commutators: the A_{2} conjecture  Carlos Pérez
 Pseudodifferential operators approach to singular integral operators
acting on weighted Lebesgue variable exponent spaces on Carleson
curves  Vladimir Rabinovich
 BMO assumptions on the coefficients and the boundary in the Dirichlet problem for higher order elliptic systems  Tatiana Shaposhnikova
 Invertibility properties of some Toeplitz operators with matrix symbols  Ilya Spitkovsky
 Weighted Fourier Inequalities  Sergey Tikhonov
 Global W^{2,p} estimate for elliptic operator with potential satisfying a reverse Hölder condition  Beatriz Viviani
 On Anisotropic Hardy Spaces  Tal Weissblat