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.

Passive vector turbulence

This thesis consists of three articles on passive vector fields in turbulence. The vector fields interact with a turbulent velocity field, which is described by the Kraichnan model. The effect of the Kraichnan model on the passive vectors is studied via an equation for the pair correlation function and its solutions. The first paper is concerned with the passive magnetohydrodynamic equations. Emphasis is placed on the so called "dynamo effect", which in the present context is understood as an unbounded growth of the pair correlation function. The exact analytical conditions for such growth are found in the cases of zero and infinite Prandtl numbers. The second paper contains an extensive study of a number of passive vector models. Emphasis is now on the properties of the (assumed) steady state, namely anomalous scaling, anisotropy and small and large scale behavior with different types of forcing or stirring. The third paper is in many ways a completion to the previous one in its study of the steady state existence problem. Conditions for the existence of the steady state are found in terms of the spatial roughness parameter of the turbulent velocity field.

Part I: Enzyme replacement versus neurotrophic factor viral vector-mediated gene therapy of Parkinson’s disease, Part II: The effects of orally dosed tolcapone and entacapone on dopamine metabolism in the dorsal striatum and nucleus accumbens in rats

Part I: Parkinson’s disease is a slowly progressive neurodegenerative disorder in which particularly the dopaminergic neurons of the substantia nigra pars compacta degenerate and die. Current conventional treatment is based on restraining symptoms but it has no effect on the progression of the disease. Gene therapy research has focused on the possibility of restoring the lost brain function by at least two means: substitution of critical enzymes needed for the synthesis of dopamine and slowing down the progression of the disease by supporting the functions of the remaining nigral dopaminergic neurons by neurotrophic factors. The striatal levels of enzymes such as tyrosine hydroxylase, dopadecarboxylase and GTP-CH1 are decreased as the disease progresses. By replacing one or all of the enzymes, dopamine levels in the striatum may be restored to normal and behavioral impairments caused by the disease may be ameliorated especially in the later stages of the disease. The neurotrophic factors glial cell derived neurotrophic factor (GDNF) and neurturin have shown to protect and restore functions of dopaminergic cell somas and terminals as well as improve behavior in animal lesion models. This therapy may be best suited at the early stages of the disease when there are more dopaminergic neurons for neurotrophic factors to reach. Viral vector-mediated gene transfer provides a tool to deliver proteins with complex structures into specific brain locations and provides long-term protein over-expression. Part II: The aim of our study was to investigate the effects of two orally dosed COMT inhibitors entacapone (10 and 30 mg/kg) and tolcapone (10 and 30 mg/kg) with a subsequent administration of a peripheral dopadecarboxylase inhibitor carbidopa (30 mg/kg) and L- dopa (30 mg/kg) on dopamine and its metabolite levels in the dorsal striatum and nucleus accumbens of freely moving rats using dual-probe in vivo microdialysis. Earlier similarly designed studies have only been conducted in the dorsal striatum. We also confirmed the result of earlier ex vivo studies regarding the effects of intraperitoneally dosed tolcapone (30 mg/kg) and entacapone (30 mg/kg) on striatal and hepatic COMT activity. The results obtained from the dorsal striatum were generally in line with earlier studies, where tolcapone tended to increase dopamine and DOPAC levels and decrease HVA levels. Entacapone tended to keep striatal dopamine and HVA levels elevated longer than in controls and also tended to elevate the levels of DOPAC. Surprisingly in the nucleus accumbens, dopamine levels after either dose of entacapone or tolcapone were not elevated. Accumbal DOPAC levels, especially in the tolcapone 30 mg/kg group, were elevated nearly to the same extent as measured in the dorsal striatum. Entacapone 10 mg/kg elevated accumbal HVA levels more than the dose of 30 mg/kg and the effect was more pronounced in the nucleus accumbens than in the dorsal striatum. This suggests that entacapone 30 mg/kg has minor central effects. Also our ex vivo study results obtained from the dorsal striatum suggest that entacapone 30 mg/kg has minor and transient central effects, even though central HVA levels were not suppressed below those of the control group in either brain area in the microdialysis study. Both entacapone and tolcapone suppressed hepatic COMT activity more than striatal COMT activity. Tolcapone was more effective than entacapone in the dorsal striatum. The differences between dopamine and its metabolite levels in the dorsal striatum and nucleus accumbens may be due to different properties of the two brain areas.

First Observation of Vector Boson Pairs in a Hadronic Final State at the Tevatron Collider

We present the first observation in hadronic collisions of the electroweak production of vector boson pairs (VV, V=W, Z) where one boson decays to a dijet final state. The data correspond to 3.5  fb-1 of integrated luminosity of pp̅ collisions at √s=1.96  TeV collected by the CDF II detector at the Fermilab Tevatron. We observe 1516±239(stat)±144(syst) diboson candidate events and measure a cross section σ(pp̅ →VV+X) of 18.0±2.8(stat)±2.4(syst)±1.1(lumi)  pb, in agreement with the expectations of the standard model.

First Observation of Vector Boson Pairs in a Hadronic Final State at the Tevatron Collider

We present the first observation in hadronic collisions of the electroweak production of vector boson pairs (VV, V=W,Z) where one boson decays to a dijet final state . The data correspond to 3.5 inverse femtobarns of integrated luminosity of ppbar collisions at sqrt(s)=1.96 TeV collected by the CDFII detector at the Fermilab Tevatron. We observe 1516+/-239(stat)+/-144(syst) diboson candidate events and measure a cross section sigma(ppbar->VV+X) of 18.0+/-2.8(stat)+/-2.4(syst)+/-1.1(lumi) pb, in agreement with the expectations of the standard model.

Search for new color-octet vector particle decaying to t tbar in p pbar collisions at sqrt{s} = 1.96 TeV

We present the result of a search for a massive color-octet vector particle, (e.g. a massive gluon) decaying to a pair of top quarks in proton-antiproton collisions with a center-of-mass energy of 1.96 TeV. This search is based on 1.9 fb$^{-1}$ of data collected using the CDF detector during Run II of the Tevatron at Fermilab. We study $t\bar{t}$ events in the lepton+jets channel with at least one $b$-tagged jet. A massive gluon is characterized by its mass, decay width, and the strength of its coupling to quarks. These parameters are determined according to the observed invariant mass distribution of top quark pairs. We set limits on the massive gluon coupling strength for masses between 400 and 800 GeV$/c^2$ and width-to-mass ratios between 0.05 and 0.50. The coupling strength of the hypothetical massive gluon to quarks is consistent with zero within the explored parameter space.

Inference on Cointegration in Vector Autoregressive Models (summary section only)

In the thesis we consider inference for cointegration in vector autoregressive (VAR) models. The thesis consists of an introduction and four papers. The first paper proposes a new test for cointegration in VAR models that is directly based on the eigenvalues of the least squares (LS) estimate of the autoregressive matrix. In the second paper we compare a small sample correction for the likelihood ratio (LR) test of cointegrating rank and the bootstrap. The simulation experiments show that the bootstrap works very well in practice and dominates the correction factor. The tests are applied to international stock prices data, and the .nite sample performance of the tests are investigated by simulating the data. The third paper studies the demand for money in Sweden 1970—2000 using the I(2) model. In the fourth paper we re-examine the evidence of cointegration between international stock prices. The paper shows that some of the previous empirical results can be explained by the small-sample bias and size distortion of Johansen’s LR tests for cointegration. In all papers we work with two data sets. The first data set is a Swedish money demand data set with observations on the money stock, the consumer price index, gross domestic product (GDP), the short-term interest rate and the long-term interest rate. The data are quarterly and the sample period is 1970(1)—2000(1). The second data set consists of month-end stock market index observations for Finland, France, Germany, Sweden, the United Kingdom and the United States from 1980(1) to 1997(2). Both data sets are typical of the sample sizes encountered in economic data, and the applications illustrate the usefulness of the models and tests discussed in the thesis.

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.