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Complex and Harmonic Analysis 2011

July, 10th - 14th, 2011
Málaga, Spain

Main How to's Participants Program

Documentation for the Meeting will be handed to the participants late on Sunday the 10th, and early on Monday the 11th. Lectures will be delivered from Monday to Thursday, leaving the afternoons of Monday, Wednesday and Thursday at the participants leisure.

PROGRAM
8:30 - 9:00
Info Desk
Monday, 11th
9:10 - 9:15 :: Welcoming
9:15 - 9:50
10:00 - 10:35
10:45 - 11:10
11:15 - 11:45
Coffee Break
11:45 - 12:20
12:30 - 13:05
13:15 - 13:50
Tuesday, 12th
9:00 - 9:35
9:45 - 10:20
10:30 - 11:05
11:15 - 11:45
Coffee Break
11:45 - 12:20
12:30 - 13:05
13:15 - 13:40
Wednesday, 13th
9:00 - 9:35
9:45 - 10:20
10:30 - 11:05
11:15 - 11:45
Coffee Break
11:45 - 12:20
12:30 - 13:05
13:15 - 13:50
Thursday, 14th
9:30 - 9:55
10:00 - 10:25
10:30 - 11:05
11:15 - 11:45
Coffee Break
11:45 - 12:20
12:30 - 13:05
...
LUNCH
...
LUNCH
_\ ˇ ˇ ! ˇ ˇ /_
ThE eNd
16:00 - 16:25
16:30 - 16:55
17:00 - 17:25
17:30 - 18:00
Coffee Break
18:00 - 18:25
18:30 - 18:55
19:00 - 19:25
19:30 - 19:55
...
DINNER
ABSTRACTS
Alexandru ALEMAN Univ. Lund, Sweeden
Alexandru.Aleman@math.lu.se Wednesday, 13th, 9:00 - 9:35
A quantitative estimate for bounded point evaluations in $P2(\mu)$-spaces
Given a compactly supported positive measure $\mu$ in the complex plane, $P2(\mu)$ is the closure of analytic polynomials in $L2(\mu)$. A description of these spaces has been obtained by J. Thomson in 1991, a famous result that considerably improves S. Brown's proof of the existence of nontrivial invariant subspaces of operators with normal extensions. The basic idea is the dichotomy: Either $P2(\mu)=L2(\mu)$, or the functional of evaluation at some fixed point extends to a bounded linear functional on $P2(\mu)$. The proof however is not constructive and does not yield information about the norm of these evaluations, or their location. The purpose of the talk is to show how the, (seemingly unrelated) work of X. Tolsa on analytic capacity can be used to address these questions.
This is joint work with S. Richter and C. Sundberg.
Luis BERNAL-GONZÁLEZ Univ. Sevilla, Spain
lbernal@us.es Tuesday, 12th, 18:00 - 18:25
Topological and algebraic structure of the set of strongly annular functions
In this talk, we consider the space $H(\mathbb D)$ of holomorphic functions in the open unit disk $\mathbb D$ of the complex plane. By an elementary application of the maximum modulus principle, there is no function $f \in H(\mathbb D )$ such that $\lim_{|z| \to 1} |f(z)| = +\infty$, or equivalently, $\lim_{r \to 1} m(r,f) = +\infty$, where $m(r,f) := \min \{|f(z)|: \, |z| = r\}$ $(0<r<1)$. But functions $f \in H(\mathbb D)$ with the weaker property \[ \limsup_{r \to 1} \,m(r,f) = +\infty \] do exist, and they are called {\em strongly annular functions}. Starting from D. Bonar (1971), these functions have been extensively studied from several points of view. Specially, D. Bonar and F. Carroll proved in 1975 that strongly annular functions occur in a generic way: they form a residual subset (=: ${\mathcal SA}$) of $H(\mathbb D)$ when this space is endowed with the compact-open topology. And recently, D. Redett (2007) has been able to construct a function $f \in {\mathcal SA}$ in each generalized Bergman space $A^P_\alpha(\mathbb D)$, defined, for $0< p <+\infty$ and $\alpha > -1$, as \[ A^p_\alpha (\mathbb D ) := \{f \in H(\mathbb D ): \,\iint_{\mathbb D} |f(z)|^p (1 - |z|)^\alpha \,dx\, dy < +\infty\}. \] In this talk, we show that the residuality can be extended to several spaces of holomorphic functions. Moreover, it is proved that the family ${\mathcal SA}$ is large not only topologically, but even algebraically: there exist dense vector subspaces as well as infinitely generated algebras contained, except for zero, in ${\mathcal SA}$. A number of open problems are proposed. [Joint work with A. Bonilla].
Dimitrios BETSAKOS Univ. Thessaloniki, Greece
betsakos@math.auth.gr Wednesday, 13th, 11:45 - 12:20
A variant of Schwarz's lemma
Suppose that $f$ is holomorphic in the unit disk $\mathbb D$ and $f(\mathbb D)\subset \mathbb D$, $f(0)=0$. A classical inequality due to Littlewood generalizes Schwarz's lemma and asserts that for every $w\in f(\mathbb D)$, we have $|w|\leq \prod_j |z_j(w)|$, where $z_j(w)$ is the sequence of pre-images of $w$. We prove a similar inequality replacing the assumption $f(\mathbb D)\subset \mathbb D$ by the weaker assumption $\text{Diam} f(\mathbb D)=2$. The main tools in the proof are Green's function and Steiner symmetrization.
Oscar BLASCO Univ. Valencia, Spain
Oscar.Blasco@uv.es Wednesday, 13th, 9:45 - 10:20
Remarks on weighted mixed norm spaces
We find conditions on a measurable function $\rho:(0,1]\to \mathbb R^+$, which is bounded on compact sets, for the boundedness of the Bergman projection, the Berezin transform and the averaging operator to hold on weighted mixed norm spaces $L(p,q,\rho)$ consisting of measurable functions satisfying \[ \left(\int_0^1 \frac{\rho(1-r)}{1-r}\Bigl(\int_0^{2\pi} |f(re^{i\theta})|^p \,\frac{d\theta}{2\pi}\Bigr)^{q/p}\,dr\right)^{1/q}<\infty. \] Our results extend those known for $L^p(\mathbb D, (1-|z|^2)^\alpha dA(z))$ and mixed norm spaces $L(p,q,\alpha)$.
José BONET Univ. Politécnica de Valencia, Spain
jbonet@mat.upv.es Wednesday, 13th, 10:30 - 11:05
Holomorphic dependence of diagonal operators between sequence spaces
In this lecture we investigate properties of diagonal operators defined on Köthe echelon spaces in case the diagonal depends holomorphically on a parameter $z \in \mathbb{D}$. To do this we study operator-weighted composition maps $W_{\psi, \varphi}: f \mapsto \psi (f \circ \varphi)$ on unweighted $H(\mathbb{D},X)$ and weighted $H_v^{\infty}(\mathbb{D},X)$ spaces of vector valued holomorphic functions on the unit disc $\mathbb{D}$. Here $\varphi$ is an analytic self-map of $\mathbb{D}$ and $\psi$ is an analytic operator-valued function on $\mathbb{D}$ and $X$ is a Fréchet space. We characterize when the operator is continuous or maps bounded sets into relatively compact sets. In this way we extend results due to Laitila and Tylli for the case of Banach valued functions. This more general setting permits us to compare the results in the unweighted and weighted case. The diagonal operators between Köthe echelon spaces show the differences between the present setting and the case of functions taking values in Banach spaces.
Antonio BONILLA Univ. La Laguna, Spain
abonilla@ull.es Tuesday, 12th, 16:30 - 16:55
Rate of growth of D-frequently hypercyclic functions
We study the possible rates of growth of frequently hypercyclic entire functions for the differentiation operator. The following diagram represents our present knowledge of possible or impossible growth rates $e^r/r^a$ for $M_p(f,r):=\bigl(\frac{1}{2\pi}\int_0^{2\pi} |f(re^{it})|^p\,dt\bigr)^{1/p}$, with $1\leq p < \infty$, and $M_\infty(f,r):=\sup_{|z|=r} |f(z)|$.
  1. Blasco, O.; Bonilla, A.; Rate of growth of frequently hypercyclic functions; Proc. Edinburgh Math Soc.; 53 (2010), pp. 39-59
  2. Bonet, J.; Bonilla, A. Chaos of the differentiation operator on weighted Banach spaces of entire functions; To appear in Complex Analysis and Operator Theory;
María J. CARRO Univ. Barcelona, Spain
carro@ub.edu Monday, 11th, 9:15 - 9:50
Weighted estimates in a limited range with applications to the Bochner-Riesz operators
In this talk, I shall present the latest result of a joint work with Javier Duoandikoetxea and María Lorente concerning a problem for the so-called limited extrapolation theory of Rubio de Francia.
This theory states that if we know the boundedness of an operator $T$ on $L^{p_0}(u)$ for a subclass of weights in $A_{p_0}$, we can deduce the boundedness of $T$ on $L^p(u)$ for every $p>1$ and every $u$ in a certain subclass of $A_p$. We shall prove that we can also deduce two kinds of weighted estimates, from $L^p(u)$ to $L^{q, \infty}(v)$ and from the weighted Lorentz space $\Lambda^p(w)$ to $\Lambda^{p, \infty}(w)$.
The applications include the Bochner-Riesz operators and some others. We also consider the results in the case of one-sided operators.
Manuel D. CONTRERAS Univ. Sevilla, Spain
contreras@us.es Tuesday, 12th, 12:30 - 13:05
Loewner Theory in Annulus
Loewner Theory in the unit disk was originated in K. Loewner's seminal paper of 1923, where he introduced the so-called parametric representation of univalent functions as a powerful tool for solving extremal problems of Geometric Function Theory, including the famous Bieberbach's conjecture. Since that time Loewner Theory has gone far beyond the scope of the original problem. The most spectacular example is a stochastic version of the Loewner equation (SLE) introduced in 2000 by O. Schramm, which appeared to be intrinsically related to several important lattice models in Statistical Physics, such as the critical Ising model. Recently Filippo Bracci, Santiago Díaz-Madrigal and the author have proposed a general approach in Loewner Theory, which brings together similar but yet independent constructions from the classical theory in the unit disk.
In 1943, Y. Komatu was able to apply Loewner's ideas to parametric representation of univalent functions in the annulus. Since then, there have been several works on Loewner Theory for multiply connected domains, but all of them consider some special cases. This talk is devoted to a general version of Loewner Theory for the annulus. It is based on two joint works [arXiv:1011.4253] and [arXiv:1105.3187] with P. Gumenyuk and S. Díaz-Madrigal. One of the main new features of Loewner Theory in the doubly (and, more generally, multiply) connected setting is that there is no autonomous analogue of the Loewner Evolution, which is a good source of intuition for the theory in the unit disk, because for multiply connected case the semigroup structure disappears: instead of self-maps of a static domain one has to consider holomorphic mappings between one-parametric family of reference domains. We will discuss the notions of Loewner chain and evolution family over an expanding system of annuli and its relationship with Carathéodory ODEs driven by semi-complete non-autonomous holomorphic vector fields.
George COSTAKIS Univ. Crete, Greeece
costakis@math.uoc.gr Monday, 11th, 13:15 - 13:50
Extreme limiting behavior of the partial sums of smooth functions
This is joint work with J. Muller. Let $A(\mathbb D)$ denote the disk algebra, i.e., the set of holomorphic functions in the open unit disk, $\mathbb D$, which extend continuously on the closure of $\mathbb D$. We investigate the behavior of the partial sums of typical functions $f\in A(\mathbb D)$ on compact sets of the unit circle. In particular we show that there exists $g\in A(\mathbb D)$ so that on certain thin subsets of the unit circle the partial sums of $g$ behave extremely wild. On the other hand, it is well known that the partial sums $S_n(g, 0)$ of every function $g\in A(\mathbb D)$ behave regular on fat subsets of the unit circle $\mathbb T$, that is, $S_n(g, 0)$ converges to $g$ a.e. on $\mathbb T$. As a byproduct of our work we give a negative answer to a question of Pichorides.
Rodrigo HERNÁNDEZ Univ. Adolfo Ibáńez, Chile
rodrigo.hernandez@uai.cl Thursday, 14th, 10:00 - 10:25
Schwarzian derivative for harmonic functions
We give a definition of schwarzian derivative for harmonic functions that preserve the orientation and no additional assumption on the complex dilatation $\omega$. The way to understand this formula is based on the fact that it should characterize the harmonic Möbius transformations. The definition characterizes the Möbius harmonic mappings as a natural way to extend the schwarzian derivative for this kinds of functions.
H. Turgay KAPTANOĞLU Univ. Bilkent, Ankara, Turkey
kaptan@fen.bilkent.edu.tr Thursday, 14th, 9:30 - 9:55
Three problems for weighted Bloch-Lipschitz spaces of holomorphic functions on the unit ball
We consider weighted Bloch-Lipschitz spaces $\mathcal B_\alpha$ of holomorphic functions on the unit ball $\mathbb B$ of $\mathbb C^N$. For any $\alpha\in\mathbb R$, they are defined by requiring that $\|f\|_\alpha :=\sup\,\{\,(1-|z|^2)^{\alpha+t}\,D^tf(z):z\in\mathbb B\,\}<\infty$ for some $t$ satisfying $\alpha+t>0$, where $D^t$ is a radial differential operator of order $t\in\mathbb R$.
We concentrate on three problems on $\mathcal B_\alpha$ that can be solved for all $\alpha\in\mathbb R$. First, for each $b\in\mathbb B$, we determine the unique extremal function realizing $\sup\,\{\,f(b)>0:f\in\mathcal B_\alpha,\;\|f\|_\alpha=1\,\}$. Second, we prove that $\mathcal B_\alpha$ contains those $\alpha$-Möbius-invariant spaces that possess a decent linear functional. Third, for $\alpha>0$, we find new complete Hermitian metrics $\rho_\alpha$ with which the functions in $\mathcal B_\alpha$ satisfy a Lipschitz-type inequality, which is well-known for $\alpha\leq0$.
This is joint work with S. Tülü.
Gabriela KOHR Univ. Babeş-Bolyai, Romania
gkohr@math.ubbcluj.ro Tuesday, 12th, 13:15 - 13:40
Extreme points, support points and univalent subordination chains in $\mathbb{C}^n$
In this talk we survey recent results in the theory of Loewner chains and the generalized Loewner differential equation in several complex variables. Also we investigate certain properties of extreme points and support points of the family $S^0(B^n)$ of normalized biholomorphic mappings on the unit ball in $\mathbb{C}^n$ that have parametric representation.
This talk is based on joint work with Mirela Kohr (Cluj-Napoca), Ian Graham (Toronto) and Hidetaka Hamada (Fukuoka).
Mirela KOHR Univ. Babeş-Bolyai, Romania
mkohr@math.ubbcluj.ro Tuesday, 12th, 19:00 - 19:25
Potential analysis for pseudodifferential Brinkman operators on Lipschitz domains
The purpose of this talk is to present a potential analysis for pseudodifferential Brinkman operators on Lipschitz domains in the Euclidean setting or in compact Riemannian manifolds. We apply this analysis to treat transmission problems for such operators.
This is a joint work with Cornel Pintea (Cluj-Napoca) and Wolfgang L. Wendland (Stuttgart).
María J. MARTÍN Univ. Autónoma Madrid, Spain
mariaj.martin@uam.es Thursday, 14th, 10:30 - 11:05
Besov spaces, multipliers, and univalent functions
An analytic function $f$ in the unit disk $\mathbb{D}$ belongs to the conformally invariant Besov space $B^p$ ($1< p<\infty$) if its derivative $f^\prime$ belongs to the weighted Bergman space $A^p_{p-2}$. The minimal space $B^1$ is the set of analytic functions $f$ in $\mathbb{D}$ such that $f^{\prime\prime}\in A^1$.
We focus on the problem of the boundedness of multiplication operators between the Besov spaces $B^p$ ($1\leq p<\infty $). We look for checkable descriptions of the spaces of multipliers $M(B^p, B^q)$ and give an extensive class of explicit examples. We also study which functions of certain important types (lacunary series, univalent functions, modified-inner functions) are to be found in the spaces $M(B^p, B^q)$.
This is a joint work with Petros Gananopoulos and Daniel Girela.
Francisco J. MARTÍN-REYES Univ. Málaga, Spain
martin_reyes@uma.es Monday, 11th, 12:30 - 13:05
A dominated ergodic theorem for some bilinear averages
Let $T$ be a positive invertible linear operator with positive inverse on some $L^p(\mu)$, $1\leq p<\infty$, where $\mu$ is a $\sigma$-finite measure. We study the convergence in the $L^p(\mu)$-norm and the almost everywhere convergence of the bilinear operators \[ \mathcal A_n(f_1,f_2)=\left(\frac{1}{2n+1}\sum_{i=-n}^nT^if_1(x)\right) \left(\frac{1}{2n+1}\sum_{i=-n}^nT^if_2(x)\right) \] for functions $f_1\in L^{p_1}(\mu)$ and $f_2\in L^{p_2}(\mu)$, $1\leq p, 1<p_1,p_2<\infty$, $1/p_1+1/p_2=1/p$. It turns out to be that the convergence in $L^p(\mu)$ is equivalent to the dominated estimate for the ergodic maximal operator associated to $\mathcal A_n$ and to the uniform boundedness of the operators $\mathcal A_n$. It is also shown that the convergence in the $L^p(\mu)$-norm implies the almost everywhere convergence. On one hand, the key facts to prove these results are transference arguments and the connection with a new class of weights recently introduced by Lerner et al. (Advances in Mathematics 220 (2009) 1222-1264). On the other hand, our main result can be viewed as the ergodic counterpart of one of the main results in the above cited paper.
Margit PAP Univ. Vienna, Austria
margit.pap@univie.ac.at Monday, 11th, 10:45 - 11:10
Multiresolution in $H^2(\mathbb T)$ generated by a special Mamquist-Takenaka system
In signal processing and system identification for $H^2(\mathbb T)$ and $H^2(\mathbb D)$ the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. In this talk we present a set of poles and using them we will generate a multiresolution in $H^2(\mathbb T)$ and $H^2(\mathbb D)$. The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space $H^2(\mathbb T)$. The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to $H^2(\mathbb T)$ and $H^2(\mathbb D)$ if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.
Christos PAPACHRISTODOULOS Univ. Crete, Greece
papach@math.uoc.gr Tuesday, 12th, 17:00 - 17:25
Universality, summability and Rogosinski's formula
Let $\Omega \neq \mathbb C$ be an arbitrary domain in $\mathbb C$ and $\zeta\in\Omega$. Let $R = {\rm dist}(\zeta, \partial\Omega)$ and $K = \partial\Omega \cap \{z \in \mathbb C:\ |z-\zeta|=R\}$. We prove the existence of universal functions $f \in {\mathcal H}(\Omega)$ for $K$, that is the sequence of partial sums of the Taylor development $\sum_{n=0}^\infty c_n (z-\zeta)^n$ of $f$ approximates uniformly any continuous function on $K$. Also we give sufficient conditions on $K$ which guarantee that the above series is not $(C, a)$-summable for every $a>-1$. Finally we give examples of universal Taylor series $\sum_{n=0}^\infty c_n z^n$ for $K$, with $K$ a finite subset of $\mathbb T=\{z\in\mathbb C:\ |z|=1\}$, such that the series $\sum_{n=0}^\infty c_n z^n$ is $(C,a)$-summable for every $a\geq 1$.
Mihalis PAPADIMITRAKIS Univ. Crete, Greece
papadim@math.uoc.gr Tuesday, 12th, 9:45 - 10:20
Composition operators on $B_1$
We review some previous results on the compactness and the essential norm of a composition operator $C_\phi$ on $B_1$ and, mainly, describe a necessary condition on the symbol $\phi$ for $C_\phi$ to have closed range.
José A. PELÁEZ Univ. Málaga, Spain
japelaez@uma.es Tuesday, 12th, 10:30 - 11:05
Spectra of integration operators on Hardy spaces
We shall present weighted versions of the classical estimates due to Fefferman-Stein and Littlewood-Paley which express the $H^p$-norm of an analytic function with help of its derivative. These results shall be used to study the spectrum of integration operators \[ T_g f(z)=\int_0^zf(\xi)g'(\xi)d\xi, \] acting on the Hardy spaces $H^p$. (Joint work with Alexandru Aleman.)
Carlos PÉREZ Univ. Sevilla, Spain
carlosperez@us.es Monday, 11th, 11:45 - 12:20
Theory of singular integrals and weights: Recent results
It is well known that the basic operators from Harmonic Analysis are bounded on weighted Lp spaces when the weight satisfies the Ap condition. There is now a growing interest in understanding the behavior of the operator norm in terms of the Ap constant of the weight. The main result is the so called the A2 theorem which was a conjecture until few months ago. We plan to discuss this theorem and a recent improvement obtained in joint work with T. Hytönen.
Fernando PÉREZ-GONZÁLEZ Univ. la Laguna, Spain
fernando.perez.gonzalez@ull.es Wednesday, 13th, 12:30 - 13:05
Inner functions in Möbius invariant spaces
The classical problem of determining which inner functions (or their derivatives) belong to a given space of analytic function has a long story. In the talk we will focus on the Möbius invariant Besov-type spaces $B^p_s$, ($p>0$, $s\geq0$), which consist of the analytic functions in the open unit disc $\mathbb D$ such that \[ \sup_{a \in \mathbb D} \int_{\mathbb D} |f'(z)|^p (1 - |z|^2)^{p-2} g^s(z,a)dA(z) < \infty,% \qquad p >0, \quad s \geq 0, \] where $g(z,a) = - \log|\varphi_a(z)|$ is the Green function of $\mathbb D$ and $\varphi_a(z) = (a-z)/(1- \overline{a}z)$. In fact, some of our results are stated for the general classes $F(p,q,s)$ ($p >0$, $ -2 < q < \infty$, $ 0 \leq s < \infty$, and $q + s> -1$), consisting of those analytic functions in $\mathbb D$ for which \[ \sup_{a \in \mathbb D} \int_{\mathbb D} |f'(z)|^p (1 - |z|^2)^{q} g^s(z,a)dA(z) < \infty. \] We prove that for $0<s\leq1$, the atomic singular inner function $S_{\gamma,w}\in F(p,q,s)$ if and only if $p\leq q+(s+3)/2$. As a consequence, it follows that $S_{\gamma, w}\not\in B^p_s$ and $S_{\gamma, w}\not\in B^p_{1,0}$, but $S_{\gamma, w}\in B^p_{1}$. This is generalized by showing that for $0<p<\infty$ and $0\leq s<1$ such that $p+s>1$, $B^p_s$ does not contain any singular inner functions.
Then, it is seen that Blaschke products belonging to $B^p_s$ are those for which their zero sequences $\{z_n\}$ sequence satisfy $\sup_{a\in\mathbb D} \sum_{n=1}^\infty (1-|\varphi_a(z_n)|^2)^s <\infty$.
(The talk is based upon a joint work with Jouni Rättyä).
Karl M. PERFEKT Univ. Lund, Sweeden
perfekt@maths.lth.se Tuesday, 12th, 18:30 - 18:55
On duality and distance results in $BMO$ type spaces
For the classical space of functions with bounded mean oscillation, it is well known that $VMO^{**} = BMO$ and there are many characterizations of the distance from a function $f$ in $BMO$ to $VMO$. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this talk it will be explained how to obtain such duality results and distance formulas by pure functional analysis. Applications include Möbius invariant spaces, Hölder spaces and rectangular $BMO$ of several variables.
Sandra POTT Univ. Lund, Sweeden
sandra@maths.lth.se Wednesday, 13th, 13:15 - 13:50
A dyadic approach to Sarason's conjecture on Toeplitz products on Bergman space
In 1994, D. Sarason posed conjectures on the characterization of the boundedness of Toeplitz products on Hardy and Bergman spaces. The Hardy space case attracted much attention because of its close relation to the famous two-weight problem for the Hilbert transform in Analysis, but was shown to be false by F. Nazarov around 2000. The Bergman space case is still open. In the talk, I will present a dyadic model with a test function criterion for the Bergman space case. This is joint work with A. Aleman.
Stamatis POULIASIS Univ. Thessaloniki, Greece
spoulias@math.auth.gr Tuesday, 12th, 16:00 - 16:25
Equality cases for condenser capacity inequalities under symmetrization
It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.
Aristomenis SISKAKIS Univ. Thessaloniki, Greece
siskakis@math.auth.gr Thursday, 14th, 12:30 - 13:05
Strong continuity of composition semigroups in spaces of analytic functions - A survey
Given a semigroup $\{\phi_t: t\geq 0\}$ of analytic self maps of the unit disc $\mathbb{D}$ there arises a composition operator semigroup \[ T_t(f)=f\circ\phi_t, \quad f\in \mathcal{H}(\mathbb{D}), \] on the space $\mathcal{H}(\mathbb{D})$ of all analytic functions on $\mathbb{D}$. If $X$ is a linear subspace of $\mathcal{H}(\mathbb{D})$ with a norm $\|\cdot\|_X$ such that the composition operators $T_t$ are bounded on $X$, we are interested when $\{T_t\}$ are strongly continuous on $X$. We will survey results on classical spaces $X$ of analytic functions, stressing the case of $BMOA$ where the question is more difficult and partially open.
Javier SORIA Univ. Barcelona, Spain
soria@ub.edu Tuesday, 12th, 9:00 - 9:35
Solution to a conjecture on the Hardy operator minus the identity and a new class of minimal rearrangement invariant spaces
N. Kruglyak and E. Setterqvist [PAMS 136 (2008), 2505--2513] have shown that the norm of the Hardy operator minus the identity $S-I$, on the cone of decreasing functions in $L^p$ and for $p\in\{2,3,\dots\}$, is given by: \[ \Vert S-I\Vert_p=\frac{1}{(p-1)^{1/p}}, \] and conjectured that the estimate would also hold for $p\geq 2$.
In collaboration with S. Boza [JFA 260 (2011), 1020--1028] we have proved that this is true, even for a larger class of weights $w$: If $w\in B_p$ (the Arińo and Muckenhoupt class) and \[ r^{p-1}\int_r^{\infty} \frac{w(x)}{x^p} \, dx \leq \frac{w(r)}{p-1}, \quad \text{a.e. $r>0$,} \] then, \[ \Vert S-I\Vert_{p,w}=\|w\|_{B_p}^{1/p},\qquad p\geq2. \] Observe that if the operator $S-I:L^p_{\rm dec}(w)\longrightarrow L^p(w)$ is bounded, necessarily $w\in B_p$.
Motivated by these results, for a rearrangement invariant space $X$ we have introduced [Studia Math. 197 (2010), 69--79] a new space $R(X)$, where \[ \|f\|_{R(X)}=\int_0^\infty W_X\bigl(\lambda_f(t)\bigr)\, dt<+\infty, \] and $W_X(t)=\| 1/{(1+\frac{\cdot}t)}\|_X$, and have characterized when $R(X)\neq \{0\}$ or $R(X)=\Lambda(X)$ (joint work with S. Rodríguez-López, to appear in Proc. Edinb. Math. Soc.). Some other new properties will also be discussed.
Georgios STYLOGIANNIS Univ. Thessaloniki, Greece
stylog@math.auth.gr Tuesday, 12th, 19:30 - 19:55
A brief review on Brennan's Conjecture
Brennan's conjecture concerns integrability of the derivative of a conformal map $g$ of a simply connected planar domain $G$ onto the unit disk $\mathbb{D}$. The conjecture is that, for all such $G$ and $g$, \[ \int_{G}|g'(z)|^{p}dA(z)<\infty \] holds for $4/3<p<4$. Here $dA$ is the area measure on the plane.
We will present a brief review of results concerning Brennan's conjecture. We will focus on the work of W. Smith and S. Shimorin that establishes a connection between the conjecture and weighted composition operators.
Ana VARGAS Univ. Autónoma Madrid, Spain
ana.vargas@uam.es Monday, 11th, 10:00 - 10:35
Restriction theorems for surfaces with vanishing curvatures
This is a joint work with Sanghyuk Lee. We prove some bilinear restriction estimates for conic type surfaces with more than one vanishing curvature. As corollary, we show new restriction estimates for some conic surfaces.
Dragan VUKOTIĆ Univ. Autónoma Madrid, Spain
dragan.vukotic@uam.es Tuesday, 12th, 11:45 - 12:20
Derivatives of Blaschke products and Bergman spaces with normal weights
This talk is based on a recent joint work with A. Aleman.
We consider the membership of the derivative of a Blaschke product in a Bergman space with normal weights (in the sense of Shields and Williams) and present several results that generalize or complement earlier findings by various authors.
Nina ZORBOSKA Univ. Manitoba, Canada
zorbosk@cc.umanitoba.ca Thursday, 14th, 11:45 - 12:20
Schwarzian derivative and general Besov-type domains
We will look at the univalent functions $f$ for which $\log f'$ belongs to a class of general Besov-type spaces. We provide a characterization of the general Besov-type domains in terms of Carleson measure conditions on the schwarzian derivative of $f$.