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.

Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Opioid dependence: Brain structure and function : a magnetic resonance imaging, neuropsychological, and electromagnetic study

Background: Opiod dependence is a chronic severe brain disorder associated with enormous health and social problems. The relapse back to opioid abuse is very high especially in early abstinence, but neuropsychological and neurophysiological deficits during opioid abuse or soon after cessation of opioids are scarcely investigated. Also the structural brain changes and their correlations with the length of opioid abuse or abuse onset age are not known. In this study the cognitive functions, neural basis of cognitive dysfunction, and brain structural changes was studied in opioid-dependent patients and in age and sex matched healthy controls. Materials and methods: All subjects participating in the study, 23 opioid dependents of whom, 15 were also benzodiazepine and five cannabis co-dependent and 18 healthy age and sex matched controls went through Structured Clinical Interviews (SCID) to obtain DSM-IV axis I and II diagnosis and to exclude psychiatric illness not related to opioid dependence or personality disorders. Simultaneous magnetoencephalography (MEG) and electroencephalography (EEG) measurements were done on 21 opioid-dependent individuals on the day of hospitalization for withdrawal therapy. The neural basis of auditory processing was studied and pre-attentive attention and sensory memory were investigated. During the withdrawal 15 opioid-dependent patients participated in neuropsychological tests, measuring fluid intelligence, attention and working memory, verbal and visual memory, and executive functions. Fifteen healthy subjects served as controls for the MEG-EEG measurements and neuropsychological assessment. The brain magnetic resonance imaging (MRI) was obtained from 17 patients after approximately two weeks abstinence, and from 17 controls. The areas of different brain structures and the absolute and relative volumes of cerebrum, cerebral white and gray matter, and cerebrospinal fluid (CSF) spaces were measured and the Sylvian fissure ratio (SFR) and bifrontal ratio were calculated. Also correlation between the cerebral measures and neuropsychological performance was done. Results: MEG-EEG measurements showed that compared to controls the opioid-dependent patients had delayed mismatch negativity (MMN) response to novel sounds in the EEG and P3am on the contralateral hemisphere to the stimulated ear in MEG. The equivalent current dipole (ECD) of N1m response was stronger in patients with benzodiazepine co-dependence than those without benzodiazepine co-dependence or controls. In early abstinence the opioid dependents performed poorer than the controls in tests measuring attention and working memory, executive function and fluid intelligence. Test results of the Culture Fair Intelligence Test (CFIT), testing fluid intelligence, and Paced Auditory Serial Addition Test (PASAT), measuring attention and working memory correlated positively with the days of abstinence. MRI measurements showed that the relative volume of CSF was significantly larger in opioid dependents, which could also be seen in visual analysis. Also Sylvian fissures, expressed by SFR were wider in patients, which correlated negatively with the age of opioid abuse onset. In controls the relative gray matter volume had a positive correlation with composite cognitive performance, but this correlation was not found in opioid dependents in early abstinence. Conclusions: Opioid dependents had wide Sylvian fissures and CSF spaces indicating frontotemporal atrophy. Dilatation of Sylvian fissures correlated with the abuse onset age. During early withdrawal cognitive performance of opioid dependents was impaired. While intoxicated the pre-attentive attention to novel stimulus was delayed and benzodiazepine co-dependence impaired sound detection. All these changes point to disturbances on frontotemporal areas.

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.

Kotitalouden opetustilat osana kehittyvää oppimisympäristöä : asumistoiminnot ja opetussuunnitelma muutoksen määrittäjinä

Home Economics Classrooms as Part of Developing the Environment Housing Activities and Curriculums Defining Change --- The aim of the research project was to develop home economics classrooms to be flexible and versatile learning environments where household activities might be practiced according to the curriculum in different social networking situations. The research is based on the socio-cultural approach, where the functionality of the learning environment is studied specifically from an interactive learning viewpoint. The social framework is a natural starting point in home economics teaching because of the group work in classrooms. The social nature of learning thus becomes a significant part of the learning process. The study considers learning as experience based, holistic and context bound. The learning environment, i.e. home economics classrooms and the material tools there, plays a significant role in developing students skills to manage everyday life. --- The first research task was to analyze the historical development of household activities. The second research task was to develop and test criteria for functional home economics classrooms in planning both the learning environment and the students activities during lessons. The third research task was to evaluate how different professionals (commissioners, planners and teachers) use the criteria as a tool. The research consists of three parts. The first contains a historical analysis of how social changes have created tension between traditional household classrooms and new activities in homes. The historical analysis is based on housing research, regulations and instructions. For this purpose a new theoretical concept, the tension arch, was introduced. This helped in recognizing and solving problems in students activities and in developing innovations. The functionality criteria for home economics classrooms were developed based on this concept. These include technical (health, safety and technical factors), functional (ergonomic, ecological, aesthetic and economic factors) and behavioural (cooperation and interaction skills and communication technologies) criteria. --- The second part discusses how the criteria were used in renovating school buildings. Empirical data was collected from two separate schools where the activities during lessons were recorded both before and after classrooms were renovated. An analysis of both environments based on video recordings was conducted. The previously created criteria were made use of, and problematic points in functionality looked for particularly from a social interactive viewpoint. The results show that the criteria were used as a planning tool. The criteria facilitated layout and equipment solutions that support both curriculum and learning in home economics classrooms taking into consideration cooperation and interaction in the classroom. With the help of the criteria the home economics classrooms changed from closed and complicated space into integrated and open spaces where the flexibility and versatility of the learning environment was emphasized. The teacher became a facilitator and counselor instead a classroom controller. --- The third part analyses the discussions in planning meetings. These were recorded and an analysis was conducted of how the criteria and research results were used in the planning process of new home economics classrooms. The planning process was multivoiced, i.e. actors from different interest groups took part. All the previously created criteria (technical, functional and behavioural) emerged in the discussions and some of them were used as planning tools. Planning meetings turned into planning studios where boundaries between organizations were ignored and the physical learning environments were developed together with experts. The planning studios resulted in multivoiced planning which showed characteristics of collaborative and participating planning as well as producing common knowledge and shared expertise. --- KEY WORDS: physical learning environment, socio-cultural approach, tension arch, boundary crossing, collaborative planning.

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.

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.