On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

Numerical approaches to new particle formation and growth in the atmosphere

Aerosols impact the planet and our daily lives through various effects, perhaps most notably those related to their climatic and health-related consequences. While there are several primary particle sources, secondary new particle formation from precursor vapors is also known to be a frequent, global phenomenon. Nevertheless, the formation mechanism of new particles, as well as the vapors participating in the process, remain a mystery. This thesis consists of studies on new particle formation specifically from the point of view of numerical modeling. A dependence of formation rate of 3 nm particles on the sulphuric acid concentration to the power of 1-2 has been observed. This suggests nucleation mechanism to be of first or second order with respect to the sulphuric acid concentration, in other words the mechanisms based on activation or kinetic collision of clusters. However, model studies have had difficulties in replicating the small exponents observed in nature. The work done in this thesis indicates that the exponents may be lowered by the participation of a co-condensing (and potentially nucleating) low-volatility organic vapor, or by increasing the assumed size of the critical clusters. On the other hand, the presented new and more accurate method for determining the exponent indicates high diurnal variability. Additionally, these studies included several semi-empirical nucleation rate parameterizations as well as a detailed investigation of the analysis used to determine the apparent particle formation rate. Due to their high proportion of the earth's surface area, oceans could potentially prove to be climatically significant sources of secondary particles. In the lack of marine observation data, new particle formation events in a coastal region were parameterized and studied. Since the formation mechanism is believed to be similar, the new parameterization was applied in a marine scenario. The work showed that marine CCN production is feasible in the presence of additional vapors contributing to particle growth. Finally, a new method to estimate concentrations of condensing organics was developed. The algorithm utilizes a Markov chain Monte Carlo method to determine the required combination of vapor concentrations by comparing a measured particle size distribution with one from an aerosol dynamics process model. The evaluation indicated excellent agreement against model data, and initial results with field data appear sound as well.

Vain hätapua? : Taloudellinen avustaminen diakoniatyön professionaalisen itseymmärryksen ilmentäjänä

Financial Help Alone? Financial help as an exponent of professional diaconal work One essential form of helping people in the Evangelical Lutheran Church s diaconal work is providing economic aid. It can be seen as work which is in accordance with the spirit of the Church Order (4:3). One of the tasks of diaconal work, determined by the Church Order, is to help those whose distress is the greatest and who have no other source of help. This financial support has become a permanent and essential working method, which has also created tension of various kinds. Financial support has been criticized, especially when the support has been used to fill a gap in the social services provided by the government. It has been argued that diaconal work has been forced to take on responsibility for tasks that belong to the welfare state. The tensions involved in the financial support of diaconal work do not only concern the patching up and supplementing of the deficiencies in the welfare state s services but also the question of diaconal workers self-understanding of financial support and how it relates to their professionalism. In this thesis, I examine the experiences and visions diaconal workers have concerning financial support in their work with clients. The viewpoint of my work is the diaconal workers own experiences and interpretations of the meaning of financial support in customer service. In the articles of my thesis, I examined the meanings that diaconal workers gave to financial support in the aspects of work motivation, empowerment, expertise and tensions. The research material of my articles consists of three different data, which are theme interviews from diaconal workers, a survey from diaconal workers of Espoo and a diaconal barometer of 2009. I have analysed the theme interviews and the survey using qualitative content analysis. The results of my articles showed that diaconal workers motivation in tasks concerning economic aid was sustained by the nature and spiritual aspects of support activities. Work that supported empowerment through financial assistance meant influencing the client s personal life, community and local ties and structural circumstances of the surrounding society. Diaconal workers expertise in financial support work can be characterised as horizontal, which means that the expertise was built on acknowledging the client s dignity, the uniqueness of the client s life situation and listening to the client s own voice. Diaconal workers were also experts in community and area-based work. The tensions in financial support work are linked to its unofficial and undefined role in the field of social welfare and the inability of other aiding parties to respond to their duties. The results of my thesis on the experiences and visions of financial support reveal that it is multilateral and multidimensional. Diaconal workers used financial support to help the clients, taking into account their individual, communal, social and spiritual context. The professionalism of this financial support is reflectively related to the client s need of help and the spontaneity and unexpectedness of the situation. Support work was deeply bound to diaconal workers experiences of spirituality as the basic value in their work, the foundation of their idea of humanity and their method of helping others. In different tasks of financial support diaconal workers balanced between traditional, individual client work based on caritas and working methods which are based on supporting the individual s empowerment and active citizenship, as in postmodern social work. Diaconal workers experiences of financial support illustrated the transition or turning point in the professionalism of diaconal work, which involves finding one s own, stronger and clearer professional identity than earlier with respect to other helpers in society. Creating a unique identity is part of the empowerment process of diaconal work, in which it must define its professional role by itself. In postmodern pluralism and the fragmented context of diaconal activities, the question arose as to whether the spiritual traditions and traditional values of diaconal work support the modifications and adaptations needed in new, unpredictable situations. Diaconal work is said to be fast to react, able to predict changes and adapt to those changes. To preserve its sensitive reactive ability, also in the complex postmodern world, it must retain its own views and orientations. Otherwise, the distinctive values and traditions of diaconal work might sustain static diaconal work, employee-centeredness and a smug attitude when defining beneficiaries and needs, which highlights the paternalism of diaconal work. Such paternalism may complicate the progress of working methods which are based on empowerment and citizenship.

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.