WS-CAOT 2016

We shall introduce the space of Bloch functions and present several characterizations of it. We shall concentrate on results on Taylor coefficients and multipliers.

We shall develop the theory on the unit disc. Starting with the definition of a Bloch function and basic examples we shall consider the space of such functions denoted by $\mathcal B$ and define a norm on it. The students should be able to show that it is a complete normed space and that the closure of the polynomials under such a norm, denoted by $\mathcal B_0$, can be described in several ways. We shall see that the space $\mathcal B$ is a dual space and identify the dual of $\mathcal B_0$. We shall present different characterizations of the space $\mathcal B$ by means of the pseudo-hyperbolic metric and using the Bergman projection. Another basic objective is to analyze the Taylor coefficient of Bloch functions and to give some results on coefficient multipliers acting on such space. We shall indicate several open problems concerning the Bloch functions when they are defined on $\mathbb C^n$ for $n\geq 2$ and even the new line recently introduced when the functions are defined on the unit ball of $\ell^2$.

Bibliography.

1. J.M. Anderson, J. Clunie and C.Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math., 270 (1974), 12-37.
2. J.M. Anderson, A.L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc., 224 (1976), 256-265.
3. K. Zhu, Operators on Bergman spaces, Marcel Decker, Inc., New York, 1990.

Let $M$ be a complex manifold and $\varphi:M\to M$ an holomorphic self-map of $M$. We can define the iterates $\varphi_1=\varphi$ and $\varphi_n=\varphi_{n-1}\circ\varphi$ for all $n\geq 2$. One of the main goals of Complex Dynamics is the study of the asymptotic behavior of the sequence of functions $\{\varphi_n\}$. The first results of this theory goes back to the XIXth century and the beginning of the XXth century. The names of famous mathematicians such as Poincaré, Fatou, Denjoy, Wolff, Carathéodory, Julia, ... quickly arise when database are consulted on this topic.

Nowadays, this topic is still very much alive with the emergence of new techniques and the possibility of connections and applications to other areas of mathematics such as operator theory, ergodic theory, probability or differential equations.

Undoubtedly, in this theory the geometric aspects play an important role and is essential the manifold where we are working on. In setting of one dimensional simply connected complex manifolds, by the Poincaré and Koebe uniformization theorem, we can reduce to study the following three very different types of domains:

• The Riemann sphere: iteration of rational functions;
• The complex plane $\mathbb C$: iteration of entire functions;
• The unit disk $\mathbb D$: iteration of self-maps of the unit disk.

The analysis of the latter is the main objective of this course. Montel's theorem clarifies the difference between the unit disk and the other two cases: If $\varphi:\mathbb D\to \mathbb D$ is analytic, then $\{\varphi_n\}$ is a normal family.

The study of the dynamics of an arbitrary analytic self-map $\varphi$ of the (open) unit disk $\mathbb{D}$ is a classical and well-established branch of complex iteration. Probably, the central result in the area is the celebrated Denjoy-Wolff theorem, which states that,

Theorem 1 (Denjoy-Wolff Theorem, 1926). If $\varphi:\mathbb D\to \mathbb D$ is analytic and is not an elliptic automorphism, then the iterates $(\varphi _{n})$ converge to a certain point $\tau\in\overline{\mathbb{D}}$ uniformly on compacta of $\mathbb{D}$.

This point is clearly unique and it is called the Denjoy-Wolff point of $\varphi$.

By Fatou's theorem, the above iterates $(\varphi _{n})$ are indeed well-defined (in the angular or radial sense) in the boundary of the unit disk $\partial\mathbb{D}$, up to a set of Lebesgue measure zero. In other words, there is a set $A\subseteq\partial\mathbb{D}$, with Lebesgue measure zero, such that the radial limit $\varphi _{n}(\xi ):=\lim_{r\to 1 }\varphi _{n}(r\xi)$ exists, for all $\xi \in \partial\mathbb{D}\setminus A$ and for all $n\in \mathbb{N}$. This fact opens the possibility of considering (almost everywhere) the iteration $(\varphi _{n})$ in the closed unit disk and poses the natural question about how the classical Denjoy-Wolff theorem can be extended to this situation. Besides this problem has been treated by several authors, a definitive boundary version of the Denjoy-Wolff theorem is still open, mainly due to the difficulties that appeared in the so-called parabolic case.

In this course, we will prove Denjoy-Wolff theorem and analyze three different techniques to tackle the boundary extension to $\partial \mathbb D$. The first tool appeared in [1] and uses results about composition operators on Hardy spaces. Bourdon, Matache and Shapiro proved:

Theorem 2 ([1], Theorem 4.1). If $\varphi:\mathbb D\to \mathbb D$ is non-inner analytic and its Denjoy-Wolff point $\tau$ belongs to $\mathbb D$, then $\varphi_n$ converges to $\tau$ a.e. on $\partial \mathbb D$.

Theorem 3 ([1], Theorem 4.2). Suppose $\varphi:\mathbb D\to \mathbb D$ is analytic and its Denjoy-Wolff point $\tau$ belongs to $\partial \mathbb D$.

1. If $\sum _n(1-|\varphi_n(0)|) \lt \infty$, then $\varphi_n$ converges to $\tau$ a.e. on $\partial \mathbb D$.
2. If $\varphi$ is inner and $\varphi_n$ converges to $\tau$ a.e. on $\partial \mathbb D$, then $\sum _n(1-|\varphi_n(0)|)<\infty$.

Corollary 4 ([1], Theorem 4.4). Suppose $\varphi:\mathbb D\to \mathbb D$ is analytic, its Denjoy-Wolff point $\tau$ belongs to $\partial \mathbb D$ and $\varphi'(\tau)\lt 1$. Then $\varphi_n$ converges to $\tau$ a.e. on $\partial \mathbb D$.

There have been two more recent papers where this problem has been tackled. Namely, one by Poggi-Corradini [3] and another by Díaz-Madrigal, Pommerenke and the author of these notes [2]. The technique used in [3] is based on the behaviour of certain harmonic measures associated with the iterates $\varphi_n$, while in [2], the authors used techniques coming from geometric function theory. It is worth mentioning that a general version of the boundary Denjoy-Wolff problem is open.

The program of the course is the following:

• Monday: A general introduction to Complex Dynamics with examples and the statement of the Denjoy-Wolff Theorem.
• Tuesday: We will give the proof the Denjoy-Wolff Theorem and analyze common tools to understand the proofs of the proof of Theorems 2 and 3 and Corollary 4.
• After these sessions, each student will deepen in one of the different tools to show a partial answer to the Boundary Denjoy-Wolff theorem.

Bibliography.

1. P.S. Bourdon, V. Matache, and J.H. Shapiro, On convergence to the Denjoy-Wolff point, Illinois J. Math. 49 (2005), 405--430.
2. M.D. Contreras, S. Díaz-Madrigal, and Ch. Pommerenke, Boundary behavior of the iterates of a self-map of the unit disk, J. Math. Anal. Appl., 396 (2012), 93--97.
3. P. Poggi-Corradini, Pointwise convergence on the boundary in the Denjoy-Wolff theorem, Rocky Mountain Journal of Mathematics, 40 (2010), 1275--1288.
4. J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.

We will explore the basic properties of functions in the classical Hardy space of analytic functions in the unit disc and prove the classical theorem of Beurling describing the shift invariant subspaces. Given an inner function, we will study the boundary behavior of functions in the corresponding model space.

This course will present a smooth introduction to the celebrated Krzyż conjecture, an extremal problem for analytic functions in the disk. The problem will serve for setting the stage for various basic techniques of one complex variable: Schwarz-Pick Lemma, finite Blaschke products and their characterization, Herglotz representation theorem, integral means and Hardy spaces, normal families, extremal problems and coefficient estimates, elementary variational methods, analytic covering maps, Carathéodory-Toeplitz theorem for analytic functions from the disk into the right half-plane, Féjer-Riesz theorem for positive trigonometric polynomials and/or others, as time permits.

The contents of a couple of selected chapters of monographs and parts of research papers on the course topic will be presented. These materials will be available to the students together with a set of notes covering parts of the course contents.

The task of each student will consist in studying the materials furnished, in understanding some exercises and basic examples in tutored sessions with the lecturer and, finally, in presenting an oral exposition of 10-15 minutes of a small part of the work.

In this talk we introduce the mixed norm spaces of Hardy type, a family of spaces given by the integrability of the integral means of analytic functions. We will see the relation with the classical Hardy and Bergman spaces and some growth properties of the functions belonging to these spaces, comparing them with their classical counterpart. Finally we will define the semigroups of composition operators and present their study on mixed norm spaces.

The second part of the talk is joint work with Manuel D. Contreras and Luis Rodríguez-Piazza from Universidad de Sevilla.

It is well known that harmonic functions satisfy the so called mean value property. Moreover, a harmonic function $u$ in $\Omega\subset\mathbb{R}^n$ can be characterized by the asymptotic mean value property, i.e., \[ u(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}u(y)\ dy+o(r^2). \] Similarly, we consider the asymptotic expansions \[ u(x)=\frac{\alpha}{2}\Big(\inf_{B(x,r)}u+\sup_{B(x,r)}u\Big)+\frac{1-\alpha}{|B(x,r)|}\int_{B(x,r)}u(y)\ dy+o(r^2), \] where $\alpha$ is a real number. In particular, for $1 < p < \infty$ and $\alpha=\frac{p-2}{n+p}$, this asymptotic mean value property is closely related to $p$-harmonic functions, which are the weak solutions of the $p$-Laplace equation \[ \Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)=0. \] In fact, this property characterizes $p$-harmonic functions directly for $n=2$, and in a viscosity sense for $n\geq 3$.

However, for $p\neq 2$, a new class of functions appears when we consider the nonlinear mean value property for some particular families of balls. These functions are called $p$-harmonius and, under certain assumptions, they approach $p$-harmonic functions as the radii go to zero.

We consider the Carathéodory class of functions which are analytic in the unit disk and have positive real part. We will present some recent work on inequalities for some homogeneous coefficient functionals in this class. We will draw a direct connection with the Krzyż conjecture.

In this talk, we will consider the Schur product for matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a Hilbert space. We shall show that this Schur product defines a bilinear continuous map on $\mathcal{B}(\ell^2(H))$, and we shall provide some sufficient conditions to become a Schur multiplier in this setting.

Let $\varphi,\psi\in A(\mathbb{D})$ with $\|\varphi\|\leq 1$. We consider the weighted composition operator $C_{\varphi,\psi}(f)=\psi(f\circ\varphi)$ defined on the Fréchet space $H(\mathbb{D})$ and in the Banach spaces $A(\mathbb{D})$ and $H^{\infty}$. We study under which conditions on the symbol $\varphi$ and in the weight $\psi$ the Cesàro means of the orbit of $C_{\varphi,\psi}(f)$ is convergent in the corresponding spaces. When this happens the operator is said to be mean ergodic, and when in addition the convergence is uniform the operator is called uniformly mean ergodic. In the relevant case of the composition operator $C_{\varphi}$, i.e., when $\psi=1$, the mean ergodicity has been classified by Bonet and Domański. We give a complete characterization in terms of the behaviour of the iterates of symbol when the operator is considered acting on $A(\mathbb{D})$ or $H^{\infty}$.

The case of the weighted composition is studied in $H(\mathbb{D})$, extending the results of Bonet and Domański.

This is a joint work with M.J. Beltrán-Menéu, M.C. Gómez-Collado and D. Jornet.

Bibliography:

1. J. Bonet and P. Domański, A note on mean ergodic composition operators on spaces of holomorphic functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105, no. 2 (2011), 389--396.
2. M.J. Beltrán Menéu, M.C. Gómez Collado, E. Jordá, and D. Jornet, Mean ergodic composition operators on Banach spaces of analytic functions, J. Funct. Anal. (2016). DOI: 10.1016/j.jfa.2016.03.003
3. M.J. Beltrán-Menéu, M.C. Gómez Collado, E. Jordá, and D. Jornet, Mean ergodicity of weighted composition operators on spaces of holomorphic functions, preprint.

I will talk about a couple of results connected with "well distributed" random points on the sphere. Our goal is to point out the role played by the (fast) decay of an integral kernel. The first is the (almost) optimal rate of convergence of the empirical measure, for the spherical ensemble, towards the surface measure. The second is the improvement of asymptotic (upper) bounds for the minimal Riesz energies.

For $0\le\alpha\le 1$ we have the Dirichlet-type spaces, \[ \mathcal D_\alpha=\bigl\{ f\in H^2 :\ \|f\|_{\mathcal D_\alpha}^2=|f(0)|^2+\int_{\mathbb D} |f^\prime(z)|^2(1-|z|^2)^\alpha\, dm(z) < \infty\bigr\} \] and the Möbius-invariant version of these spaces, \[ \mathcal Q_\alpha=\left\{ f\in H^2 :\ \sup\limits_{b\in\mathbb D}\|f\circ \varphi_b-f(b)\|_{\mathcal D_\alpha} < \infty\right\}, \] where, for $b\in\mathbb D$, $\varphi_b(z)=\frac{b-z}{1-\overline{b}z}$, $z\in\mathbb D$, is the Möbius mapping interchanging $0$ and $b$.

In this work we consider a new subclass of Dirichlet-type spaces. For $\alpha,\lambda\in(0,1)$, define \[ \mathcal D_\alpha^\lambda = \left\{f\in H^2:\ \|f\|_{\mathcal D_\alpha^\lambda}= \sup\limits_{b\in\mathbb D}\,(1-|b|^2)^{\alpha(1-\lambda)/2}\|f\circ \varphi_b-f(b)\|_{\mathcal D_\alpha} < \infty \right\}, \] which are between the above spaces, \[ \mathcal Q_\alpha\subset\mathcal D_\alpha^\lambda\subset\mathcal D_\alpha, \quad 0<\alpha<1,\ 0<\lambda<1. \] Our purpose is to show some of their properties, as can be the growth and behavior at the circle of their elements, or conditions that ensure or are needed for composition operators to be bounded.

Bibliography:

• J. Xiao, Holomorphic $Q$ classes, Lecture Notes in Math., Springer-Verlag, vol. 1767, (2001).

In the talk we survey some results which provide necessary and sufficient conditions in order to ensure that the derivative of an inner function belongs to certain function spaces. Moreover, we characterize those radial weights $\omega$ for which the Schwarz-Pick lemma applied to the derivative of any inner function in the norm of the Bergman space $A^p_{\omega}$ does not cause any essential loss of information. This approach is based on operator theory.

In this talk I will present some results obtained in collaboration with D. Li and H. Queffélec about approximation numbers of composition operators on $H^2$ of the unit disk $\bf D$. In particular I will present a spectral radius type formula relating the approximation numbers with the Green capacity of the image of $\bf D$ by the symbol. At the end I will give some extension of these results to the $d$-dimensional polydisk.

In the study of shift invariant subspaces, the notion of best approximation arises naturally. We look at the polynomials $p$ that minimize some norm $\|pf-1\|_{A^2}$, where $f \in A^2$, the Bergman space. In this talk we answer the question of which points of the complex plane can appear as zeros of polynomials $p$ for different functions $f$, by translating an extremal problem into a problem about orthogonal polynomials and Jacobi matrices.

Let $A^p_\omega$ denote the Bergman space in the unit disc $\mathbb D$ of the complex plane induced by a radial weight $\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\leq C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The tent space $T^q_s(\nu,\omega)$, $0 < q,s < \infty$, consists of those analytic functions $f$ on the unit disc such that \[ \|f\|_{T^q_s(\nu,\omega)}^q :=\int_{\mathbb D}\left(\int_{\Gamma(\zeta)}|f(z)|^s\,d\nu(z)\right)^\frac{q}s\omega(\zeta)\,dA(\zeta) <\infty. \] Here $\Gamma(\zeta)$ is a non-tangential approach region with vertex $\zeta$ in the punctured unit disc $\mathbb D\setminus\{0\}$. We characterize the positive Borel measures $\nu$ such that $A^p_\omega$ is embedded into the tent space $T^q_s(\nu,\omega)$, where $1+\frac{s}{p}-\frac{s}{q}>0$, by considering a generalized area operator. The results are provided in terms of Carleson measures for $A^p_\omega$.

This is a joint work together with J.Á. Peláez and J. Rättyä.