Familial aggregation of type 1 diabetes and diabetic nephropathy in Finland

Type 1 diabetes (T1D) is a common, multifactorial disease with strong familial clustering. In Finland, the incidence of T1D among children aged 14 years or under is the highest in the world. The increase in incidence has been approximately 2.4% per year. Although most new T1D cases are sporadic the first-degree relatives are at an increased risk of developing the same disease. This study was designed to examine the familial aggregation of T1D and one of its serious complications, diabetic nephropathy (DN). More specifically the study aimed (1) to determine the concordance rates of T1D in monozygotic (MZ) and dizygotic (DZ) twins and to estimate the relative contributions of genetic and environmental factors to the variability in liability to T1D as well as to study the age at onset of diabetes in twins; (2) to obtain long-term empirical estimates of the risk of T1D among siblings of T1D patients and the factors related to this risk, especially the effect of age at onset of diabetes in the proband and the birth cohort effect; (3) to establish if DN is aggregating in a Finnish population-based cohort of families with multiple cases of T1D, and to assess its magnitude and particularly to find out whether the risk of DN in siblings is varying according to the severity of DN in the proband and/or the age at onset of T1D: (4) to assess the recurrence risk of T1D in the offspring of a Finnish population-based cohort of patients with childhood onset T1D, and to investigate potential sex-related effects in the transmission of T1D from the diabetic parents to their offspring as well as to study whether there is a temporal trend in the incidence. The study population comprised of the Finnish Young Twin Cohort (22,650 twin pairs), a population-based cohort of patients with T1D diagnosed at the age of 17 years or earlier between 1965 and 1979 (n=5,144) and all their siblings (n=10,168) and offspring (n=5,291). A polygenic, multifactorial liability model was fitted to the twin data. Kaplan-Meier analyses were used to provide the cumulative incidence for the development of T1D and DN. Cox s proportional hazards models were fitted to the data. Poisson regression analysis was used to evaluate temporal trends in incidence. Standardized incidence ratios (SIRs) between the first-degree relatives of T1D patients and background population were determined. The twin study showed that the vast majority of affected MZ twin pairs remained discordant. Pairwise concordance for T1D was 27.3% in MZ and 3.8% in DZ twins. The probandwise concordance estimates were 42.9% and 7.4%, respectively. The model with additive genetic and individual environmental effects was the best-fitting liability model to T1D, with 88% of the phenotypic variance due to genetic factors. The second paper showed that the 50-year cumulative incidence of T1D in the siblings of diabetic probands was 6.9%. A young age at diagnosis in the probands considerably increased the risk. If the proband was diagnosed at the age of 0-4, 5-9, 10-14, 15 or more, the corresponding 40-year cumulative risks were 13.2%, 7.8%, 4.7% and 3.4%. The cumulative incidence increased with increasing birth year. However, SIR among children aged 14 years or under was approximately 12 throughout the follow-up. The third paper showed that diabetic siblings of the probands with nephropathy had a 2.3 times higher risk of DN compared with siblings of probands free of nephropathy. The presence of end stage renal disease (ESRD) in the proband increases the risk three-fold for diabetic siblings. Being diagnosed with diabetes during puberty (10-14) or a few years before (5-9) increased the susceptibility for DN in the siblings. The fourth paper revealed that of the offspring of male probands, 7.8% were affected by the age of 20 compared with 5.3% of the offspring of female probands. Offspring of fathers with T1D have 1.7 times greater risk to be affected with T1D than the offspring of mothers with T1D. The excess risk in the offspring of male fathers manifested itself through the higher risk the younger the father was when diagnosed with T1D. Young age at onset of diabetes in fathers increased the risk of T1D greatly in the offspring, but no such pattern was seen in the offspring of diabetic mothers. The SIR among offspring aged 14 years or under remained fairly constant throughout the follow-up, approximately 10. The present study has provided new knowledge on T1D recurrence risk in the first-degree relatives and the risk factors modifying the risk. Twin data demonstrated high genetic liability for T1D and increased heritability. The vast majority of affected MZ twin pairs, however, remain discordant for T1D. This study confirmed the drastic impact of the young age at onset of diabetes in the probands on the increased risk of T1D in the first-degree relatives. The only exception was the absence of this pattern in the offspring of T1D mothers. Both the sibling and the offspring recurrence risk studies revealed dynamic changes in the cumulative incidence of T1D in the first-degree relatives. SIRs among the first-degree relatives of T1D patients seems to remain fairly constant. The study demonstrates that the penetrance of the susceptibility genes for T1D may be low, although strongly influenced by the environmental factors. Presence of familial aggregation of DN was confirmed for the first time in a population-based study. Although the majority of the sibling pairs with T1D were discordant for DN, its presence in one sibling doubles and presence of ESRD triples the risk of DN in the other diabetic sibling. An encouraging observation was that although the proportion of children to be diagnosed with T1D at the age of 4 or under is increasing, they seem to have a decreased risk of DN or at least delayed onset.

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.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean 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.

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.