Earlier seminars, 2010

• 25 November, Zohra Farnana
Title: The double obstacle problem in metric spaces

Abstract: In this seminar I will start by defining Sobolev type spaces in metric spaces and give some of their useful properties. Then I will present a special case of the double obstacle problem which is the p-harmonic functions (obstacle problem without obstacles!). In the other part of the seminar I will take about the double obstacle problem in metric spaces and present some of the obtained results in this topic.

• 11 November, Anders Björn, MAI
Title: Newtonian (Sobolev) spaces and $p$-harmonic functions on metric spaces

Abstract: Since we now have a number of people at MAI working with analysis on metric spaces, I thought it would be useful to discuss some of the basics. The talk is meant as an introduction to the area for those who are not working in it. I will discuss how one can define (first-order) Sobolev spaces on metric spaces, in particular the Hajlasz and Newtonian approaches and some pros and cons for them. I will also discuss what $p$-harmonic functions are both on ${\bf R}^n$ and metric spaces, and maybe also the more general quasiminimizers. $p$-harmonic functions and quasiminimizers have been the main theme of Jana Björn's and my own research, esp. the Dirichlet (boundary value) problem and boundary regularity.

• 28 October, Per Enflo
Title: Geometric properties of orbits of linear operators

Abstract: If T is a linear operator on a Hilbert space H (or on some Banach space) and x is an element in the space, the orbit of x under T is the sequence x,Tx,T^2 x, ... . In recent years there has been a growing interest in studying orbits of operators and their geometric properties. The problem, whether every operator on Hilbert space has an orbit, whose closed linear span is not the whole space, is obviously just a reformulation of The Invariant Subspace Problem. There is already an extensive literature on hypercyclic orbits, i.e. orbits whose closure are equal to the whole space. Some years ago it was proved that all separable Banach spaces have operators with such orbits. But for many questions the study of orbits is still in its beginning. Some examples of such questions are: How do the norms of the elements in an orbit vary? What are the angles between different elements in an orbit? I will give results and open problems for these type of questions for some familiar operators. There are many open problems of this type even for the operator T, Tf = F, where F(0)=0, F' = f, on the Hilbert space of square-integrable functions on the unit interval.

• 14 October, Lukáš Malý, MAI
Title: Introduction to Rearrangement Invariant Spaces
Abstract: Although the Lebesgue $L^p$ spaces play an essential role in mathematical analysis, there are other interesting classes of Banach spaces of measurable functions. Having briefly discussed abstract theory of Banach function spaces, the seminar will be devoted to so-called rearrangement invariant spaces where all equimeasurable functions have equal norm. Due to Luxemburg representation theorem these function spaces can be studied in the context of interval $(0, \infty)$ with 1-dimensional Lebesgue measure. The elementary maximal function $f^{**}$ and Hardy-Littlewood-Pólya relation give us a powerful tool for studying function norms in a single r. i. space. On the other hand, the concept of fundamental function allows us to compare spaces with each other.

• 7 October, Alan McIntosh, Australian National University, Canberra, Australia
Title: Potential maps for de Rham complexes on Lipschitz domains
Abstract: On a domain which is starlike with respect to a ball, integral operators related to the classical Poincaré path integral serve as potential operators for de Rham complexes without boundary conditions, and the dual class generalizing Bogovskii-type operators work for de Rham complexes with full Dirichlet boundary conditions. Such operators were introduced by Mitrea, and further studied by Mitrea, Mitrea and Monniaux. In joint work with Costabel, we prove that these operators are pseudodifferential operators of order -1, and thus obtain further regularity results for these de-Rham complexes, for example in Hardy spaces. In recent work with Costabel and Taggart, we turn our attention to unbounded special Lipschitz domains, and adapt operators constructed by Chang, Krantz and Stein to construct potential maps for de Rham complexes in this case, thus obtaining similar regularity results.

• 16 September, Natan Kruglyak, MAI
Title: Riesz's rising sun lemma, Calderón-Zygmund type decompositions and stable near-minimizers
Abstract: I plan to discuss the rising sun lemma, the classical Calderón-Zygmund decomposition and their applications, and then explain how covering theorems can b e used to construct an analog of the Calderón-Zygmund decomposition for spaces of differentiable functions.

• 9 September, Natan Kruglyak, MAI
Title: Classical covering theorems and theorem on controlled coverings
Abstract: Covering lemmas (theorems) are an interesting tool in analysis. I plan to discuss classical covering theorems (Whitney, Besicovitch, Wiener) and a theorem on controlled coverings, which was used to obtain a rather general formula for the K-functional.

• 2 September, Natan Kruglyak, MAI
Title: K-divisibility and real interpolation
Abstract: I plan to a) discuss history and give a short review of classical theory of real interpolation; b) formulate theorem on K-divisibility and show some applications.