# The Analysis Seminar in Thessaloniki 2007-2008

ARISTOTLE UNIVERSITY OF THESSALONIKI
DEPARTMENT OF MATHEMATICS
DIVISION OF MATHEMATICAL ANALYSIS
Tel. 2310 997906, Fax 2310 997945

Organizing Committee:
D.Betsakos
I. Gasparis

Seminar Room: 3rd floor in the building of the School of Sciences

# Programm of Talks

The seminar takes place every Tuesday 11-12 a.m. (unless otherwise stated)

### Tuesday, October 30, 2007.

Vasileios Spentzos, Aristotle University of Thessaloniki.
Title: The spectral theorem for self-adjoint operators on Hilbert spaces.
(Master thesis).

### Tuesday, November 6, 2007.

Nina Snigireva, University of St Andrews, Scotland.
Title: Fourier transforms of in-homogeneous self-similar measures.

### Tuesday, November 13, 2007.

Anestis Fotiadis, University of Warwick, U.K.
Title: Harmonic maps between manifolds of negative curvature.

### Tuesday, November 20, 2007.

Ioanna Kirezi, University of Crete.
Title: Monotonicity properties of Darboux sums.

## Analysis Mini Conference

### Tuesday, November 27, 2007.

organized by M. Marias

10.30-11.15 Nikos Bournaveas (Univ. of Edinbourgh),
Velocity averaging lemmas for kinetic transport equations.

11.30-12.00 Michel Marias (A.U.Th.),
Spectral multipliers and Riesz transforms on manifolds.

12.00-13.00. Coffee break

13.15-14.00 Paschalis Karageorgis (Univ. of Dublin),
Existence and behaviour of solutions for wave equations with potential.

14.15-14.45 Dimitrios Betsakos (A.U.Th.),
Symmetrization and heat kernels for the Laplacian..

### Tuesday, December 11, 2007.

Anestis Tsomidis, Aristotle University of Thessaloniki.
Title: Iteration of complex polynomials-Julia sets.
(Master thesis).

### Tuesday, December 18, 2007.

Minos Petrakis, Technical University of Crete.
Title: Radon-Nikodym property on Banach spaces.

### Tuesday, January 8, 2008.

Dimitris Koukoulopoulos, University of Illinois at Urbana-Champaign.
Title: A probabilistic approach to number theory and its application to a problem on localization of factorizations.
Abstract: Probabilistic number theory was born in 1917 in a paper by Hardy and Ramanujan and was further developed by Turan, Kubilius, Cramer, Erd\"os, Brun, Selberg and various others. Despite the deterministic nature of number theory, probabilistic arguments offer very accurate predictions about the behavior of various arithmetic quantities, which then we have to establish rigorously. I will present the application of such ideas to the following problem: how many integers up to $x$ are divisible by a product of $k$ integers with each of the factors lying in a prescribed interval?

### Tuesday, March 18, 2008.

Michel Marias, A.U.Th..
Title: The spectrum of the Laplacian and the Riesz transforms on hyperbolic manifolds.

### Tuesday, April 8, 2008.

Vassilios Rothos , Division of Mathematics, Faculty of Engineering, A.U.Th..
Title: Stationary and Travelling Waves in Nonlinear Lattices with Analytical Methods.

Abstract: We study stationary and travelling waves on a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a periodic nonlinear substrate potential. We show the existence of both uniform sliding states and periodic travelling waves as well in a two-dimensional sine-Gordon lattice equation using topological and variational methods. We present some recent results on dicrete Nonlinear Schrodinger equation with applications. We use variational methods (mountain-pass) and dynamical systems methods (center manifold, normal forms).

### Tuesday, April 15, 2008.

Athanasia Baharoglou, Department of Mathematics, A.U.Th.
Title: Extended Universal Taylor series on doubly connected domains

Abstract: We prove the existence of universal Taylor series on doubly connected domains which are valid on a part of the boundary. On another part of the boundary the universal functions and their derivatives are continuously extendable. Furthermore, we show similar results as those mentioned above with respect to every center. In conclusion, we mention that universal functions may vanish at 1.

### Tuesday, May 27, 2008.

11:00-12:00
Anna Savvopoulou, S.U.N.Y. at Albany
Title: Almost everywhere convergence of a case of weighted averages

Abstract: Given a probability measure $\ds \mu$ on $\ds\mathbb{Z}$, Calder\'on and Bellow proved a weak type inequality for the maximal operator of $\ds \mu_{n}\left(f(x)\right)=\sum_{k\in\mathbb{Z}}\mu^{n}(k)f\left(\sigma^{k}(x)\right)$ where $\ds \mu^{n}$ denotes the convolution product. This talk will focus on the case of a sequence of probability measures on $\ds \mathbb{Z}$, denoted by $\ds \left\{\mu_{n}\right\}$, obtained inductively in the following way, $\ds \mu_{1}=\nu_{1}$, $\ds \mu_{2}=\nu_{1}\ast\nu_{2}$, $\ds\ldots$, $\ds \mu_{n}=\nu_{1}\ast\nu_{2}\ast\cdots\ast\nu_{n}$, where each one of the $\ds \nu_{i}$ is in turn a strictly aperiodic probability measure on $\ds \mathbb{Z}$ with expectation $0$ and finite second moment. We will discuss the almost everywhere convergence of the operators $\ds \mu_{n}f(x)=\sum_{k\in\mathbb{Z}}\mu_{n}(k)f(\sigma^{k}x)$ for $\ds f\in L^{1}(X)$ and $\ds x\in X$. Throughout the talk $\ds \sigma$ will stand for a measure preserving transformation of a probability measure space $\ds X$.

12:00-13:00
Mihalis Kolountzakis, University of Crete
Title: The discrepancy of a needle on a checkerboard

Abstract:Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a miltiplicative constant. For the corresponding "finite" problem (NXN checkerboard) we also prove that we can color it in such a way that the above quantity is at most C \sqrt{N \log N}, for any placement of the line segment.

### Tuesday, June 3, 2008.

Christopher Wedrychowicz, S.U.N.Y. at Albany
Title: Discrepancy in the behavior of Ergodic averages along subsequences

Abstract:The classical theorem of Birkhoff states that the $\ds T^{N}f(x)=\frac{1}{N}\sum_{k=0}^{N-1}f\left(\sigma^{k}x\right)$ converges almost everywhere for $\ds x\in X$ and $\ds f\in L^{1}(X)$, where $\ds \sigma$ is a measure preserving transformation of a probability measure space $\ds X$. It was shown that there are operators of the form $\ds T^{N}f(x)=\frac{1}{N}\sum_{k=0}^{N-1}f\left(\sigma^{n_{k}}x\right)$ for a subsequence $\ds \{n_{k}\}$ of the positive integers that converge in some $\ds L^{p}$ spaces while diverging in others. The topic of this talk will examine this phenomenon in the class of Orlisz spaces $\ds \left\{L\mbox{Log}^{\beta}L:\beta>0\right\}$

### Friday, June 13, 2008.

11:00-12:00
Vassili Nestoridis, University of Athens
Title: Some recent results on universal series

Abstract:We do not know explicitly any universal Taylor series, but we show that it is possible to build an efficient algorithm constructing such a series in the disk, half-plane, a polygon or an angle. This does not seem to be possible when the boundary is complicated. Using the recently published “ Abstract theory of universal series ” we give a sufficient condition assuring existence of universal series with small (but not summable) coefficients. As applications we have universal approximation by translations of meromorphic functions, as -function, by translations of fundamental solutions of elliptic operators, by translations of approximate identities, as normal distributions, as well as, the existence of non periodic universal trigonometric series.

### Tuesday, June 24, 2008.

Nadia Gal, University of Missouri, Columbia
Title: Examples of Isometric Equivalence Problem

Abstract:Two operators A and B defined from a Banach space X into a Banach space Y are isometrically equivalent if there exist surjective isometries U_X, U_Y of X, respectively of Y, such that U_Y A = B U_X . My focus is the isometric equivalence problem for a spectrum of operators on a variety of Banach spaces. I establish conditions of the intertwining surjective isometries U_X and U_Y for matrix operators on sequence spaces, differentiated composition operators, integrated composition operators on spaces of analytic functions and integral operators on a space of absolutely continuous vector-valued functions. I prove that the Cesaro operator Cf(x)= 1/x \int_0^x f(t) dt, defined on a rearrangement-invariant Banach function space X commutes with an invertible isometry if and only if the isometry is modulus one multiple of the identity.