Past Functional Analysis Seminar

28 November 2017
17:00
Abstract

Our long term plan is to develop a unified approach to prove decomposition theorems in different structures. In our anti-dual pair setting, it would be useful to have a tool which is analogous to the so-called Schur complementation. To this aim, I will present a suitable generalization of the classical known Krein - von Neumann extension.

  • Functional Analysis Seminar
14 November 2017
17:00
Gregory Debruyne
Abstract

The classical Ingham-Karamata Tauberian theorem has many applications in different fields of mathematics, varying from number theory to $C_0$-semigroup theory and is considered to be one of the most important Tauberian theorems. We will discuss how to obtain remainder estimates in the theorem if one strengthens the assumptions on the Laplace transform. Moreover, we will give new (re­mainder) versions of this theorem under the more general one-sided Tauberian condition of $\rho(x) \ge −f(x)$ where $f$ is an arbitrary function satisfying some regularity assumptions. The talk is based on collaborative work with Jasson Vindas.

  • Functional Analysis Seminar
10 October 2017
17:00
Jonas Azzam
Abstract

For reasonable domains $\Omega\subseteq\mathbb{R}^{d+
1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.
 

  • Functional Analysis Seminar
6 June 2017
17:00
Julio Delgado
Abstract

We present some recent results on the study of Schatten-von Neumann properties for
 operators on compact manifolds. We will explain the  point of view of kernels and full symbols. In both cases

one relies on a suitable Discrete Fourier analysis depending on the domain.

We will also discuss about operators on $L^p$ spaces by using the notion of nuclear operator in the sense of

Grothendieck and deduce Grothendieck-Lidskii trace formulas in terms of the matrix-symbol. We present examples

 for fractional powers of differential operators.  (Joint work with Michael Ruzhansky)

  • Functional Analysis Seminar

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