Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Tila tilassa : Liturgian ja tilan dialogi alttarin äärellä

This study examines the organisation and transformation of altar space in the modern Evangelical Lutheran Church of Finland in liturgical and architectural perspective. The research data consists of 65 altar spaces in The Finnish Evangelical Lutheran church buildings. All of these were characterised in Church Government records as churches , built 1962 1999 and had been consecrated. The main data was collected by means of observation, photographing, and drawing sketches of altar spaces. The focus of this study concerns the organisation of modern Finnish Evangelical Lutheran altar spaces and, in particular, their changes also in relation to the liturgical movement. The challenge of this approach was especially in discovering the spatial identity of an altar space in terms of unequivocal boundaries. The analysis was realised in three stages. Interiors, the organisation of altar space, as well as architectonic qualities of altar spaces in terms of floor elevations, shapes of ceilings, lighting, and openings in the altar space were analysed. Moreover, attention was focused on furnishing and fixed versus movable pieces of furniture (such as the altar, altar rail, the pulpit, the baptismal font, and lectern). Finally, the potential qualitative and quantitative changes in altar space were examined. All in all, the majority of churches in the data featured elongated church halls with an altar at the end of the nave. To look at the data in chronological perspective, increasingly wide church halls had been built since the 1980s (yet there was only one central hall in which the altar was placed at the middle point of the church). Every third church altar was movable. As for the focal point of this study and the altar in particular, it was my aim to pay attention to the versus populum altar and its development in relation to the (Lutheran) liturgy. Hence, it was meaningful to determine, in terms of interior design, whether liturgists were able to celebrate facing the people attending the service. In the 1960s and 70s, a versus orientem altar featured in more than half of all new Finnish Lutheran churches, yet in 2000 two out of three churches featured a versus populum altar. For architectural and esthetic reasons (and not primarily due to liturgical ideas), also altars standing freely off the walls had been constructed. In terms of the liturgy, versus populum altars had been realised in expectation of increased communication between liturgist and worshippers. However, the analysis indicated that the altar could also become a divider of space. This aspect is a novel finding in relation to earlier and concurrent discussions concerning the liturgical movement. This study concluded, all in all, that altars had been increasingly constructed closer and closer to the worshiping parish and, accordingly, used increasingly often in the versus populum manner. Lecterns were often movable until the millennium this was the case in most altar spaces. Baptismal fonts did not have a permanent place in this data, and the data even included altar spaces with no baptismal fonts in the choir, nor the church hall. The position and status of fonts was generally weakened even if baptism in the Lutheran Church was regarded as one of the two sacraments together with the eucharist. The study concluded that even if baptism is regarded as a sacrament in the church, the position and status of baptismal fonts had weakened overall in newer church architecture. In other words, the tendency of the liturgical movement to emphasise the service and its celebration had obviously had its effect on the placement of baptismal fonts in the church hall. This research indicated that the pieces of furniture that mostly involved (many kinds of) visual and spatial changes included the altar and the lectern. In certain instances, fixed furnishings had been substituted by movable pieces or, moreover, new pieces of furniture and paraphernalia such as music instruments, pieces of art, tables, chairs and plants were brought in. In the Evangelical Lutheran Church of Finland, liturgical changes were principally inspired by the Catholic Church, in which liturgical changes are essentially based on Canon Law. Unlike Finnish Lutheranism, Catholicism provides detailed rules and principles even regarding the design of an altar space. According to this study, in the Finnish Lutheran Church, the primarily functional nature of given guidelines and instructions characterises several practical solutions in furnishing.

Pietismin kuva - ikkuna yksityiseen hengellisyyteen : Johann Arndtin opetukset pietistisen embleemin perustana

The image of Pietism a window to personal spirituality. The teachings of Johann Arndt as the basis of Pietist emblems The Pietist effect on spiritual images has to be scrutinised as a continuum initiating from the teachings of Johann Arndt who created a protestant iconography that defended the status of pictures and images as the foundation of divine revelation. Pietist artworks reveal Arndtian part of secret, eternal world, and God. Even though modern scholars do not regarded him as a founding father of Pietism anymore, his works have been essential for the development of iconography, and the themes of the Pietist images are linked with his works. For Arndt, the starting point is in the affecting love for Christ who suffered for the humankind. The reading experience is personal and the words point directly at the reader and thus appear as evidence of the guilt of the reader as well as of the love of God. Arndt uses bounteous and descriptive language which has partially affected promoting and picturing of many themes. Like Arndt, Philipp Jakob Spener also emphasised the heart that believes. The Pietist movement was born to oppose detached faith and the lack of the Holy Ghost. Christians touched by the teachings of Arndt and Spener began to create images out of metaphors presented by Arndt. As those people were part of the intelligentsia, it was natural that the fashionable emblematics of the 17th century was moulded for the personal needs. For Arndt, the human heart is manifested as a symbol of soul, personal faith or unbelief as well as an allegory of the burning love for Jesus. Due to this fact, heart emblems were gradually widely used and linked with the love of Christ. In the Nordic countries, the introduction of emblems emanated from the gentry s connections to the Central Europe where emblems were exploited in order to decorate books, artefacts, interiors, and buildings as well as visual/literal trademarks of the intelligentsia. Emblematic paintings in the churches of the castles of Venngarn (1665) and Läckö (1668), owned by Magnus Gabriel De la Gardie, are one of the most central interior paintings preserved in the Nordic countries, and they emphasise personal righteous life. Nonetheless, it was the books by Arndt and the Poet s Society in Nurnberg that bound the Swedish gentry and the scholars of the Pietist movement together. The Finnish gentry had no castles or castle churches so they supported county churches, both in building and in maintenance. As the churches were not private, their iconography could not be private either. Instead, people used Pietist symbols such as Agnus Dei, Cor ardens, an open book, beams, king David, frankincense, wood themes and Virtues. In the Pietist images made for public spaces, the attention is focused on pedagogical, metaphorical, and meaningful presentation as well as concealed statements.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.