Análisis Complejo y Teoría de Operadores Complex Analysis ...campus.usal.es/~rsme2011/doc/abstracts/sesion... · Congreso de la Real Sociedad Matemática Española Ávila, Febrero

Centennial Congress of the Spanish Royal Mathematical Society Ávila, February 1–5, 2011

S3

Análisis Complejo y Teoría de Operadores

Complex Analysis and Operator Theory

hrs.: 12

email: [email protected]

Organizadores / Organizers:

• Alexandru Aleman (University of Lund)

• Daniel Girela (Universidad de Málaga)

• Dragan Vukotic (Universidad Autónoma de Madrid)

Conferencias / Talks:

Tue 1, 17:30 - 18:00, SC – Konstantin Dyakonov (ICREA/Universidad de Barcelona):A Diophantine equation and zeros of analytic functions.

Tue 1, 18:00 - 18:30, SC – Jordi Pau (Universidad de Barcelona):Hankel operators on weighted Bergman spaces.

Tue 1, 18:30 - 19:00, SC – José Ángel Peláez (Universidad de Málaga):Square functions in weighted Hardy spaces.

Tue 1, 19:00 - 19:30, SC – Artur Nicolau (Universidad Autónoma de Barcelona):Differentiability of functions in the Zygmund class.

Tue 1, 19:30 - 20:00, SC – Óscar Blasco (Universidad de Valencia):Toeplitz operators with Berezin transform in mixed norm spaces.

Wed 2, 15:30 - 16:00, SC – Alexander Vasiliev (University of Bergen, Norway):Parametrization of the Loewner-Kufarev evolution in Sato’s Grassmannian.

Wed 2, 16:00 - 16:30, SC – Manuel Contreras (Universidad de Sevilla):Loewner Theory in the unit disk.

Wed 2, 16:30 - 17:00, SC – Pavel Gumenyuk (University of Bergen, Universidad deSevilla):Loewner Theory in annulus.

Wed 2, 17:00 - 17:30, SC – Arturo Fernández Arias (UNED):On the deficiencies of meromorphic functions of finite order in C-0.

Wed 2, 18:00 - 18:30, SC – Juan Bès (Bowling Green State University, OH, USA):Disjoint hypercyclic linear fractional composition operators.

Wed 2, 18:30 - 19:00, SC – Eva A. Gallardo Gutiérrez (Universidad Complutense deMadrid):A generalization of the Aleksandrov operator and adjoints of weighted compositionoperators.

Wed 2, 19:00 - 19:30, SC – Domingo García (Universidad de Valencia):Bohr’s strips for Dirichlet series in Banach spaces.

Wed 2, 19:30 - 20:00, SC – José Bonet (Universidad Politécnica de Valencia):Superposition operators on weighted Banach spaces of type H∞.

Thu 3, 18:00 - 18:30, SC – Marcus Carlsson (Lund University, Sweden):Finite interval convolution operators and a Beurling’s theorem for L2((-1,1))?.

Thu 3, 18:30 - 19:00, SC – Eugenia Malinnikova (Norwegian University of Science andTechnology, Trondheim, Norway):Radial behavior of harmonic functions.

Thu 3, 19:00 - 19:30, SC – Anders Olofsson (Lund University, Sweden):On the shift semigroup on the Hardy space of Dirichlet series.

Thu 3, 19:30 - 20:00, SC – Yurii Lyubarskii (Norwegian University of Science andTechnology, Trondheim, Norway):Sampling near the critical density.

Fri 4, 18:00 - 18:30, SC – Yacin Ameur (Lulea University of Technology, Sweden):On the Coulomb plasma.

Fri 4, 18:30 - 19:00, SC – Rodrigo Hernández (Universidad Adolfo Ibáñez, Chile):Nehari class in several complex variables.

Fri 4, 19:00 - 19:30, SC – Eva Tourís (Universidad Autónoma de Madrid):Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces.

Fri 4, 19:30 - 20:00, SC – María J. Martín (Universidad Autónoma de Madrid):Besov spaces, multipliers, and univalent functions.

Fri 4 20:00 - 20.30 SC – Cristóbal González (Universidad de Málaga):Inner functions in Besov spaces.

Congreso de la Real Sociedad Matemática EspañolaÁvila, Febrero 1–5, 2011

On the Coulomb plasmaYacin Ameur

Abstract: We are considering a gas or “plasma” of electrons in a plane, in thepresence of an external field which is strong enough near infinity that the particles beconfined to a finite portion of it. If we let the number of particles increase indefinitely,the gas will condensate on a certain compact subset of the plane, known as the“droplet”. The shape of this droplet depends on the external field. The problem todetermine the details of it is the Laplacian growth problem from fluid mechanics.This is what gives the classical equilibrium distribution of the plasma. Looking atthe gas in further detail, the first thing to observe is that the repulsion between theelectrons will cause a “very uniform distribution” in the vicinity of the droplet. Oneof our theorems assert that the fluctuations about the equilibrium converges to aGaussian field on the droplet with free boundary conditions. If there is time, I willalso mention a new kind of field approximations which we can use to justify much ofthe physical formalism of conformal field theory. Joint work with Nikolai Makarov,Håkan Hedenmaln and Nam-Guy Kang, in different constellations.

Department of Mathematics,LuleåUniversity of Technology,SE-971 87 Luleå, [email protected]


Disjoint hypercyclic linear fractional compositionoperators

Juan Bès1, Ozgür Martin2, and Alfredo Peris3

A seminal article in 1997 by Bourdon and Shapiro [3] completely characterized thecyclic and hypercyclic behaviour of a linear fractional composition operator on theHardy space. This was later thoroughly extended by Gallardo-Gutiérrez and Montes-Rodríguez [4] to weighted Dirichlet spaces. More recently, notions of disjointnessamong finitely many operators acting on the same Fréchet space were introduced[1], [2]. The aim of this talk is to provide an update on these notions, and in particu-lar a characterization of disjoint hypercyclicity among finitely many linear fractionalcomposition operators acting on spaces of holomorphic functions on the unit disk.

Keywords: Hypercyclic operators, composition operators

Mathematics Subject Classification 2000: 47A16, 47B33

Referencias[1] L. Bernal-González. Disjoint hypercyclic operators. Studia Math. 182(2),

113–131, 2007.

[2] J. Bès and A. Peris. Disjointness in hypercyclicity. J. Math. Anal. Appl. 336(1),297–315, 2007.

[3] P. S. Bourdon and J. H. Shapiro. Cyclic phenomena for composition opera-tors. Mem. Amer. Math. Soc. 125, 1997.

[4] E. A. Gallardo-Gutiérrez and A. Montes-Rodríguez. The role of thespectrum in the cyclic behaviour of composition operators. Mem. Amer. Math.Soc. 167(791), 2004.

1Department of Mathematics and Statistics,Bowling Green State University,Bowling Green, OH 43403, [email protected]

2Department of MathematicsUniversity of MiamiOxford, OH 45056, [email protected]

3Departmento de Matemática AplicadaUniversidad Politécnica de ValenciaCamino de Vera, s/n. 46022 Valencia, [email protected]


Toeplitz operators with Berezin transform in mixednorm spaces

Óscar Blasco 1 and Salvador Pérez-Esteva2

Let A2 be the Bergman space in the unit disk D. It is known that for a nonegativeϕ ∈ L1(D), the Toeplitz operator

Tϕ(f)(z) =

∫

D

f(w)ϕ(w)

(1− wz)2 dA(w) (1)

belongs to the Schatten class Sp if and only if the Berezin transform ϕ ∈ Lp(D, dλ),where dλ = dA

(1−|z|2)2 and dA is the normalized Lebesgue measure on D. The secondauthor introduced the so called Schatten-Herz classes Sp,q of all Toeplitz operatorsTϕ such that when decomposed as Tϕ =

∑∞n=1 Tϕn , with ϕn = χAnϕ and An = {z ∈

D, 1− 2−n ≤ |z| ≤ 1− 2−n−1} they satisfy(∑∞

n=1 ‖Tϕn‖qSp

)1/q<∞, proving that for

nonegative symbols Tϕ ∈ Sp,q is equivalent to the Berezin transform ϕ belonging tothe Herz space Kp,−2/pq .

In this talk we consider classes of compact operators T on A2 with T =∑∞n=1 T∆n,

with ∆n(f) =∑j∈In ajz

j where In = [2n−1, 2n) ∩ N which satisfy

( ∞∑

n=1

‖T∆n‖qSp

)1/q

<∞

and relate with those such that Berezin transform T belonging to the mixed normspaces Lp,q,α defined by the condition

‖f‖Lp,q,α =(∫ 1

0

(1− r)qα−1(

∫ 2π

0

|f(reiθ|p dθ2π

)q/pdr)1/q

<∞.

Keywords: Toeplitz operators, mixed norm spaces, Berezin transform

Mathematics Subject Classification 2000: 47B35 , 46E30

1Departamento de Análisis MatemáticoUniversidad de Valencia46100 Burjassot, Valencia, [email protected]

2Instituto de Matemáticas Unidad CuernavacaUniversidad Nacional Autónoma de MéxicoA.P. 273-3 ADMON 3, Cuernavaca, Morelos, CP 62220, Mé[email protected]


Superposition operators on weighted Banach spacesof type H∞

José Bonet

We report on joint work with Dragan Vukotic (Universidad Autónoma de Madrid,Spain).

A superposition operator Sϕ, whose symbol is a fixed entire function ϕ, is obtainedby composing ϕ ◦ f with different functions f in a given space of analytic functions.The basic question is as follows: when does an operator defined in this way mapsa given space of analytic functions into another space of analytic functions? Otherquestions about such (typically non-linear) operators are: when are they continuous,map bounded sets into bounded sets or map bounded sets into relatively compact sets?The answers are usually formulated in terms of growth of the symbol. In this lecturewe consider the case of weighted Banach spaces of analytic functions on the unit discof the complex plane defined by means of sup-norms. Complete characterizations areobtained for polynomial, logarithmic and exponential weights.

Keywords: Superposition operators, weighted, entire functions, weighted Banachspaces of holomorphic functions.

Mathematics Subject Classification 2000: 46E15,47B38.

Instituto Universitario de Matemática Pura y Aplicada IUMPAUniversidad Politécnica de ValenciaC. de Vera, E-46071 Valencia, [email protected]


Finite interval convolution operators and aBeurling’s theorem for L2((−1, 1))?

Marcus Carlsson

Let Φ : (−1, 1) → C be a function and consider the operator ΓΦ : L2((0, 1)) →L2((0, 1)) given by

ΓΦ(F )(x) =

∫ 1

0

Φ(x− y)F (y)dy, 0 < x < 1.

It goes under different names like “finite interval convolution operator”, “truncatedWiener-Hopf operator” or “Toeplitz operator on the Paley Wiener space”. We consi-der norm estimates and show that it shares many properties with Hankel operators,in particular we provide a Nehari type norm characterization, as well as norm andcompactness characterization in terms of the sequential BMO-space.

Questions concerning relationship between kernel and symbol has led me to aconjecture which could be called “a finite interval version of Beurling’s theorem”. Injoint work with C. Sundberg, we can prove that this conjecture holds for a densesubset, but in general it is open. I will present the conjecture and our partial results.

Keywords: TruncatedWiener Hopf operators, orthogonal decompositions of L2([0, 1])

Mathematics Subject Classification 2000: 46E22, 47B35

Mathematics DepartmentUniversidad de Santiago de [email protected]


Loewner Theory in the unit diskManuel D. Contreras

In 1923 K. Loewner introduced a new method in order to attack the famousBieberbach conjecture about Taylor coefficient estimates for univalent functions. Sincethen his method, known now as Parametric Representation of univalent functions, hasbeen developed by a number of prominent analysts, among which we have to mentionfundamental contributions of P.P. Kufarev and Chr. Pommerenke. The ParametricRepresentation Method, being a key ingredient in the proof of Bieberbach’s conjecture,given by L. de Brange in 1984, has however gone far beyond the scope of the initialproblem. Two spectacular examples of its applications in other areas of Mathematicsare the Loewner-Kufarev type equation describing free boundary flow of a viscous fluidin a Hele-Shaw cell; and the highly celebrated Stochastic Loewner Evolution (SLE)introduced in 2000 by O. Schramm as a powerful tool that led to deep results in themathematical theory of some 2D lattice models of great importance for StatisticalPhysics.

Originally Loewner Theory was developed for univalent functions normalized at aninternal point of the reference domain. This case is referred nowadays to as the radialLoewner Evolution. The first paper on Parametric Method for functions normalized ata boundary point was probably published in 1949 by N.V. Popova. This variant of thetheory, marked in the modern terminology by the attribute chordal, was developed byP.P. Kufarev, I.A. Aleksandrov, V.V. Sobolev, V.V. Goryainov and others. However,the importance of the chordal variant of Loewner Theory remained underestimateduntil O. Schramm, who used it to define SLE.

Another mathematical construction closely related to Loewner Theory is one-parametric semigroups of holomorphic functions. This construction provides a set ofimportant non-trivial examples for Operator Theory.

In this talk, we will present a series of recent results by F. Bracci, S. Díaz-Madrigal,P. Gumenyuk, and the author where it has been introduced a new unifying approachto Loewner Theory containing radial and chordal variants as special cases as well asone-parametric semigroups as an autonomous reduction. According to this approachthe essence of Loewner Theory can be represented via connection and interaction ofthree classes of objects:

Time-dependent Herglotz vector fields in the unit disk, which generate non-autonomous holomorphic flows in the unit disk.

Evolution families, which can be characterized as systems of holomorphic endo-morphisms of the unit disk generated by the above mentioned flows.

Loewner chains, which are one-parametric families of univalent functions withexpanding systems of image domains.

The relationship between these these three objects is: any Loewner chain satisfies alinear PDE driven by a Herglotz vector field, and the corresponding evolution familyarises by solving the characteristic equation for this PDE.

Keywords: Loewner Theory, Evolution Family, Herglotz vector filed, Loewner chain.

Mathematics Subject Classification 2000: 30C80, 30D05, 34M15

Referencias[1] M. Abate, F. Bracci, M.D. Contreras, and S. Díaz-Madrigal, The evolution of

Loewner’s differential equation. To appear in Newsletter of the European Mathe-matical Society, December 2010. Available on http://personal.us.es/contreras/1-ems-abate-etal.pdf

[2] F. Bracci, M.D. Contreras, and S. Díaz-Madrigal, Evolution Families and theLoewner Equation I: the unit disk, To appear Journal für die reine und ange-wandte Mathematik (Crelle’s Journal).

[3] F. Bracci, M.D. Contreras, and S. Díaz-Madrigal, Evolution Families and theLoewner Equation II: complex hyperbolic manifolds. Math. Ann., 344 (2009), 947–962.

[4] M.D. Contreras, S. Díaz-Madrigal, and P. Gumenyuk, Loewner chains in the unitdisk, Revista Matemática Iberoamericana, 26 (2010), 975-1012.

[5] M.D. Contreras, S. Díaz-Madrigal, and P. Gumenyuk, Geometry behind chordalLoewner chains, Complex Analysis and Operator Theory, 4 (2010), 541-587.

[6] J.B. Conway, Functions of one complex variable, II. Second edition. GraduateTexts in Mathematics, 159. Springer-Verlag, New York-Berlin, 1996.

[7] P.L. Duren, Univalent Functions, Springer, New York, 1983.

[8] P.P. Kufarev, On one-parameter families of analytic functions (in Russian. Englishsummary), Rec. Math. [Mat. Sbornik] N.S. 13 (55) (1943), 87–118.

[9] Ch. Pommerenke, Über dis subordination analytischer funktionen, J. Reine AngewMath. 218 (1965), 159–173.

[10] Ch. Pommerenke, Univalent Functions. With a chapter on quadratic differentialsby Gerd Jensen, Vandenhoeck & Ruprecht, Göttingen, 1975.

Departamento de Matemática Aplicada IIUniversidad de SevillaCamino de los Descubrimientos, s/n, 41092. [email protected]


A Diophantine equation and zeros of analyticfunctions

Konstantin Dyakonov

The so-called abc theorem for polynomials (also known as Mason’s, or Mason–Stothers’, theorem) deals with nontrivial polynomial solutions to the Diophantineequation a + b = c. It provides a lower bound for the number of distinct zeros ofthe polynomial abc in terms of the degrees of a, b and c. We prove some “local” abctype theorems for general analytic functions that live on a (reasonably nice) boundeddomain rather than on the whole plane. The estimates obtained are sharp, for anydomain, and they imply (a generalization of) the original “global” abc theorem by alimiting argument.

Keywords: Mason’s theorem, analytic functions, Blaschke products

Mathematics Subject Classification 2000: 30D50, 30D55, 11D41

ICREA & Universitat de BarcelonaDepartament de Matemàtica Aplicada i AnàlisiGran Via 585E-08007 [email protected]


On the deficiencies of meromorphic functions offinite order in C \ {0}

Arturo Fernández Arias

We shall consider meromorphic functions f : C \ {0} → C and make use of Ne-vanlinna theory in annuli as presented by R. Korhonen and A. Ya. Kristiyanin andA. A. Kondratyuk. Many of the results in the plane hold also in C \ {0} . Here weshall consider a well-known result of A. Pfluger on the deficiencies of meromorphicfunctions of finite integral order q. It is proved that this result also holds in C \ {0} ,with the further assumption of unequal growth at zero and infinity.

Keywords: Deficiencies, meromorphic function, value distribution

Mathematics Subject Classification 2000: 30D35

Departamento de Matemáticas Fundamentales,UNED. 28040 Madrid, [email protected]


A generalization of the Aleksandrov operator andadjoints of weighted composition operators

Eva A. Gallardo Gutiérrez

A generalization of the Aleksandrov operator is provided, in order to representthe adjoint of a weighted composition operator on H2 by means of an integral withrespect to a measure. In particular, we show the existence of a family of measureswhich represents the adjoint of a weighted composition operator and discuss notonly uniqueness but also the generalization of Aleksandrov–Clark measures whichcorresponds to the unweighted case, that is, to the adjoint of composition operators.(Joint work with Professor Jonathan R. Partington).

Keywords: Aleksandrov operator, Aleksandrov–Clark measures, Weighted compo-sition operators

Mathematics Subject Classification 2000: 47B33, 30D55

Referencias[1] E. A. Gallardo-Gutiérrez and J.R. Partington. A generalization of the

Aleksandrov operator and adjoints of weighted composition operators. Annales del’Institut Fourier (en prensa).

Departamento de Análisis MatemáticoFacultad de Ciencias MatemáticasUniversidad Complutense de MadridPlaza de Ciencias, 3Ciudad Universitaria28040 - [email protected]


Bohr’s strips for Dirichlet series in Banach spacesDomingo García

Bohr showed that the width of the strip (in the complex plane) on which a givenDirichlet series

∑an/n

s, s ∈ C, converges uniformly but not absolutely, is at most1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that fora given infinite dimensional Banach space Y the width of Bohr’s strip for a Dirichletseries with coefficients an in Y is bounded by 1 − 1/Cot(Y), where Cot(Y) denotesthe optimal cotype of Y . This estimate even turns out to be optimal.

The material of this talk is based on joint research with A. Defant, M. Maestreand D. Pérez-García.

Keywords: Dirichlet series, Power series, Polynomials, Banach spaces

Mathematics Subject Classification 2000: 32A05, 46B07, 46B09, 46G20

Departamento de Análisis MatemáticoUniversity of ValenciaDoctor Moliner, 5046100 Burjasot (Valencia)[email protected]


Inner functions in Besov spacesCristóbal González

This is a joint work with D. Girela y M. Jevtić. We consider the Besov spacesBp,qα , 0 < p, q ≤ ∞, 0 ≤ α < ∞. When q = ∞, they are the Lipschitz spacesΛp,α, consisting of those functions f , analytic in the unit disk D, for which ‖f‖Λp,α :=sup0<r<1(1−r)Mp(r,D

1+αf) <∞. When q <∞, the sup-norm is replaced by an Lq-norm, ‖f‖Bp,qα :=

(∫ 1

0(1− r)q−1Mq

p (r,D1+αf)dr)1/q. Our main concern with regards

to these spaces is about the inner functions they contain. Our results will complementor extend previous works on the subject by different authors.

For instance, when α ≥ 1/p, the spaces Bp,qα are really short of inner functions.Roughly speaking, the only inner functions they contain are the finite Blaschke pro-ducts, the exceptions being Bp,∞1/p with 0 < p <∞, and B∞,q0 with q > 2. Our assertionfor the case B∞,q0 follows from the fact that these spaces are included in VMOA ifand only if q ≤ 2. Moreover, when α ≥ 1/(2p), the idea is that the spaces Bp,qα onlycontain Blaschke products as inner functions, except in the cases B∞,q0 with q > 2,and Bp,∞1/(2p). In addition to these results, we give a characterization of inner functionsI in Bp,qα when max{0, 1/p− 1} < α < 1/p in terms of the distribution of the preima-ges I−1(a), a ∈ D. Finally, we study which of these spaces are natural containers forBlaschke products with zeros in a given Stolz angle.

Keywords: Blaschke products, Inner functions, Lipschitz spaces, Besov spaces,Hardy-Sobolev spaces

Mathematics Subject Classification 2000: 30D50, 30D55

Departamento de Análisis Matemático,Facultad de Ciencias,Universidad de Málaga,29071 Málaga, [email protected]


Loewner Theory in annulus

Manuel D. Contreras1, Santiago Díaz-Madrigal1 and Pavel Gumenyuk2

Keywords: Loewner Theory, Parametric Method, univalent function, evolution fa-mily, Loewner chain, annulus

Mathematics Subject Classification 2000: 30C80, 30D05, 30C35, 34M15

Loewner Theory in the unit disk originated in the seminal paper [1] by Loewner,where he introduced the so-called Parametric Representation of univalent functions asa powerful tool for solving extremal problems of Geometric Function Theory. A moregeneral form of this method, due to Pommerenke [2, 3] and Kufarev [4], led in 1984to the proof [5] of the famous Bieberbach Conjecture.

A great impact to the development of Loewner Theory was given by a discoveryof Schramm [6], who introduced the so-called Stochastic Loewner Evolution (SLE).It appears that (a variant of) the Loewner –Kufarev equation driven by the Brow-nian motion is intrinsically related to several important lattice models in StatisticalPhysics, such as critical Ising model.

Loewner Theory involves the study of the infinitesimal structure of the semi-group Hol(D,D) of all holomorphic self-maps of the unit disk D and its subgroups.In [7, 8, 9] a general approach was developed bringing together various types of (de-terministic) Loewner Evolution, corresponding to different subgroups of Hol(D,D).According to this approach, the core of modern Loewner Theory in D resides in co-rrespondence and interplay between the following three basic notions: Herglotz vectorfields G : D × [0,+∞) → C, which are semicomplete time-dependent holomorphicvector fields in the unit disk D; evolution families (ϕs,t)t≥s≥0, which can be cha-racterized as non-autonomous holomorphic semiflows generated by Herglotz vectorfields; and Loewner chains (ft)t≥0, which are one-parametric families of univalentfunctions ft : D→ C with expanding systems of image domains ft(D) satisfying acertain condition on regularity in the time parameter t. There is an essentially 1-to-1correspondence between these three classes of objects.

This talk is devoted to a general version of Loewner Theory for the annulus. One ofthe main new features of Loewner Theory for doubly (and, more generally, multiply)connected case is that one has to consider a family of canonical domains, while in thesimply connected case the canonical domain (the unit disk D) is static. Accordingly,there is no autonomous analogue of the Loewner Evolution, which is a good sourceof intuition in the simply connected case.

2Supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Me-

chanism. Supported and coordinated by Universidad Complutense de Madrid and by Instituto deMatemáticas de la Universidad de Sevilla.

In the talk we introduce the three basic notions of Loewner Theory in the doublyconnected setting and establish an essentially 1-to-1 correspondence between them,analogous to that in the simply connected case. As a special case, we obtain theGoluzin –Komatu equation [10, 11] and its generalization proposed by Lebedev [12].

The interest to Loewner Theory in the annulus is connected, in particular, to theattempts to extend the notion of SLE to doubly and multiply connected case [13, 14].

Referencias[1] K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheits-

kreises (in German). Math. Ann. 89, 103–121, 1923.

[2] Ch. Pommerenke. Über die subordination analytischer funktionen (in Ger-man). J. Reine Angew Math. 218, 159–173, 1965.

[3] Ch. Pommerenke. Univalent functions. With a chapter on quadratic differen-tials by Gerd Jensen. Vandenhoeck & Ruprecht, Göttingen, 1975.

[4] P.P. Kufarev. On one-parameter families of analytic functions (in Russian.English summary) Rec. Math. [Mat. Sbornik] N.S. 13 (55), 87–118, 1943.

[5] L. de Branges. A proof of the Bieberbach conjecture. Acta Math. 154(1-2),137–152, 1985.

[6] O. Schramm. Scaling limits of loop-erased random walks and uniform spanningtrees. Israel J. Math. 118, 221–288, 2000.

[7] F. Bracci, M.D. Contreras, and S. Díaz-Madrigal. Evolution Familiesand the Loewner Equation I: the unit disk. To appear in J. Reine Angew. Math.Available on arXiv 0807.1594

[8] F. Bracci, M.D. Contreras, and S. Díaz-Madrigal. Evolution Familiesand the Loewner Equation II: complex hyperbolic manifolds. Math. Ann. 344,947–962, 2009.

[9] M.D. Contreras, S. Díaz-Madrigal, and P. Gumenyuk. Loewner chainsin the unit disk. To appear in Revista Matemática Iberoamericana 26, 975–1012,2010. Availiable on arXiv:0902.3116

[10] Y. Komatu. Untersuchungen über konforme Abbildung von zweifach zusammen-hängenden Gebieten (in German). Proc. Phys.-Math. Soc. Japan (3) 25, 1–42,1943. Avaliable via Journal@rchive, http://www.journalarchive.jst.go.jp

[11] G.M. Goluzin. On the parametric representation of functions univalent in aring (in Russian). Mat. Sbornik N.S. 29(71), 469–476, 1951.

[12] N.A. Lebedev. On parametric representation of functions regular and univalentin a ring (in Russian). Dokl. Akad. Nauk SSSR (N.S.) 103, 767–768, 1955.

[13] D. Zhan. Stochastic Loewner evolution in doubly connected domains. Probab.Theory Related Fields 129(3), 340–380, 2004.

[14] R.O. Bauer and R.M. Friedrich. On chordal and bilateral SLE in multiplyconnected domains. Math. Z. 258 (2), 241–265, 2008.

1Departamento de Matemática Aplicada IIEscuela Técnica Superior de IngenieríaUniversidad de Sevilla,Sevilla, 41092, [email protected]@us.es

2Department of MathematicsUniversity of Bergen,Johannes Brunsgate 12,Bergen 5008, Norway

andDepartamento de Matemática Aplicada IIEscuela Técnica Superior de IngenieríaUniversidad de Sevilla,Sevilla, 41092, [email protected]


Nehari class in several complex variablesRodrigo Hernández,

In this work we construct an explicit homeomorphic extension to Cn∪{∞} onto it-self for a class of biholomorphic mappings F defined in the unit ball Bn. The extensionis a multidimensional analogue of the Ahlfors-Weill construction derived originally in1962 for certain univalent mappings of the disk. The classes considered are defined interms of the Schwarzian derivative, in our setting using a definition due to T. Odawhich generalizes the classical operator in one variable. Oda’s operator splits intodifferential operators SF and S0F of order two and three, respectively, a feature alsopresent in other formulations of the Schwarzian in several complex variables

Keywords: Schwarzian derivative, homeomorphic extension, ball, univalence, con-vexity, Bergman metric.

Mathematics Subject Classification 2000: 32H02, 32A17, 30C45.

Referencias[1] Chuaqui, M. and Osgood, B. , 1993, Sharp distortion theorems associated with

the Schwarzian derivative. Jour. London Math.Soc., 2(48), 289-298.

[2] Chuaqui, M. and Osgood, B. , 1994, Ahlfors-Weill extensions of conformal map-pings and critical points of the Poincaré metric. Comment.Math.Helv., 69, 659-668.

[3] Molzon R. and Tamanoi H.,2002, Generalized Schwarzians in several variablesand Möbius invariant differential operators. Forum Math., 14, 165-188.

[4] Nehari,Z., 1949, The Schwarzian derivative and schlicht functions.Bull.Amer.Math.Soc., 55, 545-551.

[5] Oda,T., 1974, On Schwarzian Derivatives in several variables(in Japanese). Kok-yuroku of R.I.M., Kioto Univ., 226,

[6] Yoshida,M., 1976, Canonical forms of some system of linear partial differentialequations, Proc. Japan Acad., 52, 473-476.

Facultad de Ingeniería y CienciasUniversidad Adolfo Ibáñez,Av. Balmaceda 1625, Recreo, Vña del Mar, [email protected]


Sampling near the critical densityYurii Lyubarskii

We consider Gabor frames, generated by a Gaussian function and describe beha-vior of the frame constant as the density of the lattice approaches the critical value.

This is a joint work with A. Borichev and K. Groechenig.

Keywords: Gabor frame; Frame bounds; Sampling inequality; Balian-Low theorem;Fock space; Atomization techniques

Mathematics Subject Classification 2000: 42C15 , 33C90, 94A12

Department of Mathematical SciencesNorwegian University of Sciences and TechnologyA.Getz v.1, 7491, [email protected]


Radial behavior of harmonic functionsEugenia Malinnikova

We consider classes of harmonic functions in the unit disk or unit ball whichadmit a radial majorant and show that such functions may grow or decay as fastas the majorant only along a set of radii of measure zero. For the case when themajorant fulfills a doubling condition, we give precise estimates of these exceptionalsets in terms of the Hausdorff measures. Further we obtain a version of the law ofiterated logarithm that governs radial oscillation of harmonic functions in the unitdisk which admit two-sided Korenblum estimate. The talk is based on joint workswith A. Borichev, Yu. Lyubarskii, P. Thomas and K. Eikrem.

Department of Mathematics,Norwegian University of Science and Technology,7491, Trondheim, [email protected]


Besov spaces, multipliers, and univalent functionsPetros Galanopoulos1, Daniel Girela1, and María J. Martín2

We focus on the basic problem of the boundedness of multiplication operatorsbetween the conformally invariant Besov spaces of analytic functions Bp (1 ≤ p <∞).We look for checkable descriptions of the spaces of multipliers M(Bp, Bq) and givean extensive class of explicit examples. We also study which functions of certainimportant types (lacunary series, univalent functions, “modified”-inner functions) areto be found in the spaces M(Bp, Bq).

Keywords: Besov spaces, Bloch space, Möbius invariant spaces, univalent fun-ctions, multipliers

Mathematics Subject Classification 2000: 30C35, 30H25, 47B38

1Departamento de Análisis Matemático, Universidad de Málaga.Campus de Teatinos, 29071 Málaga, Spain.galanopoulos−[email protected]@uma.es

2Departamento de Matemáticas, Facultad de Ciencias (Módulo 17),Universidad Autónoma de Madrid.29049 Madrid, [email protected]


Differentiability of functions in the Zygmund classArtur Nicolau

The Zygmund class is the natural substitute of the Lipschitz class in differentcontexts. N. Makarov proved that the set of points where a function in the Zygmundclass of the real line has bounded divided differences has Hausdorff dimension one.The situation in higher dimensions will be discussed in the talk. The new results arejoint work with Juan Jesus Donaire and Jose Gonzalez Llorente.

Keywords: Zygmund class, differentiability, Hausdorff dimension

Mathematics Subject Classification 2000: 26B05

Departament de MatemàtiquesUniversitat Autònoma de Barcelona08193 Bellaterra, Barcelona, [email protected]


On the shift semigroup on the Hardy space ofDirichlet seriesAnders Olofsson

We consider the Hardy space H2 of Dirichlet series

f(s) =

+∞∑

n=1

ann−s, <(s) > 1/2,

with finite norm

‖f‖2H2 =

+∞∑

n=1

|an|2 < +∞.

The space H2 was introduced by Hedenmalm, Lindqvist and Seip in their in 1997paper as a Dirichlet series counterpart of the standard Hardy space of the unit disc.

For every positive integer n ∈ Z+ we have a natural operator S(n) acting on H2

given by multiplication by the Dirichlet monomial n−s, that is,

S(n)f(s) = n−sf(s), <(s) > 1/2,

for f ∈ H2. This provides us with a function S : Z+ 3 n 7→ S(n) which is easily seento be a multiplicative semigroup of isometries. We characterize this shift semigroupS : Z+ → L(H2) up to unitary equivalence by means of a Wold decomposition. Asan application we have that a shift invariant subspace of H2 is unitarily equivalent toH2 if and only if it has the form ϕH2 for some H2-inner function ϕ.

Keywords: shift semigroup, Hardy space of Dirichlet series, Wold decomposition,invariant subspace

Mathematics Subject Classification 2000: 47A15, 30Bxx, 46E22

Referencias[1] Anders Olofsson. On the shift semigroup on the Hardy space of Dirichlet series.

Acta Math. Hungar. 128(3), 265–286, 2010.

Anders OlofssonLund UniversityMathematics, Faculty of ScienceP.O. Box 118S-221 00 Lund, [email protected]


Hankel operators on weighted Bergman spacesJordi Pau

We study boundedness, compactness and membership in Schatten-Von Newmanideals of Hankel operators on weighted Bergman spaces. We consider standard weights,and also rapidly decreasing weights.

Keywords: Hankel operators, Bergman spaces, Schatten classes

Mathematics Subject Classification 2000: 47B35, 47B10, 30H05

Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaGran Via 58508007 Barcelona, [email protected]


Square functions in weighted Hardy spacesJosé Ángel Peláez

We shall present a weighted version of the classical formulas due to Fefferman-Stein and Littlewood-Paley which express the Hp-norm of a function with the helpof the derivative. These results shall be used to study the spectrum a class of integraloperators.

Joint work with Alexandru Aleman.

Keywords: Square functions, Integral operators.

Mathematics Subject Classification 2000: 30H10, 47G10

Referencias[1] C. Fefferman and E. M. Stein. Hp spaces of several variables. Acta Math

129 no. 3-4, 137–193, (1972).

[2] A. Aleman and O. Constantin. Spectra of integration operators on weightedBergman spaces. J. Anal. Math., 109, 199-231, (2009).

Departamento de Análisis MatemáticoUniversidad de MálagaCampus de Teatinos29071 Má[email protected]


Graphs and Gromov hyperbolicity of non-constantnegatively curved surfaces

Eva Tourís

We are interested in studying conditions which determine when a given completeRiemanian surface S is Gromov hyperbolic. In order to do it, the main goal of thiswork is to get graph-structures G, which are good models for surfaces and, in this way,moving the study of Gromov hyperbolicity from the surface to its associated graph,whose structure is very much simpler and, therefore, to study Rips condition shall beeasier.

Gromov hyperbolicity is of quite interest in metric graphs theory since it is closelyrelated to concepts arising in the study of trees: in fact, we can consider hyperbolicgraphs as a generalization of metric trees.

More precisely, in this work we obtain the equivalence of the Gromov hyperbolicitybetween an extensive class of complete Riemannian surfaces with pinched negativecurvature and certain kind of simple graphs, whose edges have length 1, constructedfollowing an easy triangular design of geodesics in the surface.

Keywords: Gromov hyperbolicity, Gromov hyperbolic graph, Hyperbolic geometry,Riemannian surface, pinched negative curvature.

Mathematics Subject Classification 2000: 53C20, 53C21, 53C22, 53C23.

Referencias[1] Chavel, I., Feldman, E.A., Cylinders on surfaces, Comment. Math. Helvetici 53

(1978), 439-447.

[2] Portilla, A., Rodríguez, J. M., Tourís, E., Structure Theorem for Riemanniansurfaces with arbitrary curvature. Preprint.

[3] Portilla, A., Tourís, E., A characterization of Gromov hyperbolicity of surfaceswith variable negative curvature, Publ. Mat. 53 (2009), 83–110.

[4] Rodríguez, J. M., Tourís, E., Gromov hyperbolicity of Riemann surfaces. ActaMath. Sinica. 23 (2007).

Departamento de MatemáticasUniversidad Carlos III de MadridAvenida de la Universidad 3028911 Leganés, [email protected]


Parametrization of the Loewner-Kufarev evolution inSato’s Grassmannian

Alexander Vasiliev

We discuss complex and Cauchy-Riemann structures of the Virasoro algebra andof the Virasoro-Bott group in relation with the Loewner-Kufarev evolution. Basedon the Hamiltonian formulation of this evolution we obtain an infinite number ofconserved quantities and provide embedding of the Loewner-Kufarev evolution intoSato’s Grassmannian. This is a joint work with Irina Markina (University of Bergen).

Department of Mathematics,University of Bergen,Johannes Brunsgate 12, N-5008 Bergen,[email protected]

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