Santiago Díaz-Madrigal

Evolution families and the Loewner equation I: the unit disc

Abstract In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson–Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.

Loewner chains in the unit disk

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains

We characterize infinitesimal generators of semigroups of holomorphic self-maps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.

Second angular derivatives and parabolic iteration in the unit disk

In this paper we deal with second angular derivatives at Denjoy-Wolff points for parabolic functions in the unit disc. Namely, we study and analyze the existence and the dynamical meaning of this second angular derivative. For instance, we provide several characterizations of that existence in terms of the so-called Koenigs function. It is worth pointing out that there are two quite different classes of parabolic iteration: those with positive hyperbolic step and those with zero hyperbolic step. In the first case, the Koenigs function is in the Caratheodory class but, in the second case, it is even unknown if it is normal. Therefore, the ideas and techniques to approach these two cases are really different. In the end, we also present several rigidity results related to the second angular derivatives at Denjoy-Wolff points.

Loewner theory in annulus I: Evolution families and differential equations

Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s Stochastic Loewner Evolution (SLE). Recently Bracci et al [10, 11, 16] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. We construct an analogue of this theory for the annulus. In this paper, the first of two articles, we introduce a general notion of an evolution family over a system of annuli and prove that there is a 1-to-1 correspondence between such families and semicomplete weak holomorphic vector fields. Moreover, in the non-degenerate case, we establish a constructive characterization of these vector fields analogous to the nonautonomous Berkson–Porta representation of Herglotz vector fields in the unit disk [10].

Classical and Stochastic Löwner–Kufarev Equations

In this paper we present a historical and scientific account of the development of the theory of the Lowner–Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Lowner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory.

Fractional iteration in the disk algebra: prime ends and composition operators

In this paper we characterize the semigroups of analytic functions in the unit disk which lead to semigroups of operators in the disk algebra. These characterizations involve analytic as well as geometric aspects of the iterates and they are strongly related to the classical theorem of Caratheodory about local connection and boundary behaviour of univalent functions

Ranges of vector measures in Fréchet spaces

Abstract Characterizations are given of those Frechet spaces E such that every compact subset of E lies in the range of an E-valued measure of bounded variation, respectively in the range of a measure of bounded variation with values in a superspace of E. Extending results for Banach spaces due to Pineiro and Rodriguez-Piazza, we prove that this property characterizes nuclear spaces, respectively hilbertizable spaces, in the framework of Frechet spaces.

Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like BMOA, Qα, and analytic Besov spaces Bp. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding “small” spaces we also characterize the boundedness and show that it is equivalent to compactness.

Semigroups of holomorphic functions in the polydisk

In this paper we provide an easy-to-use characterization of infinitesimal generators of semigroups of holomorphic functions in the polydisk. We also present a number of examples related to that characterization.