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.

Endemic malaria: an 'indoor' disease in northern Europe

Background: Endemic northern malaria reached 68°N latitude in Europe during the 19th century, where the summer mean temperature only irregularly exceeded 16°C, the lower limit needed for sporogony of Plasmodium vivax. Because of the available historical material and little use of quinine, Finland was suitable for an analysis of endemic malaria and temperature. Methods: Annual malaria death frequencies during 1800–1870 extracted from parish records were analysed against long-term temperature records in Finland, Russia and Sweden. Supporting data from 1750–1799 were used in the interpretation of the results. The life cycle and behaviour of the anopheline mosquitoes were interpreted according to the literature. Results: Malaria frequencies correlated strongly with the mean temperature of June and July of the preceding summer, corresponding to larval development of the vector. Hatching of imagoes peaks in the middle of August, when the temperature most years is too low for the sporogony of Plasmodium. After mating some of the females hibernate in human dwellings. If the female gets gametocytes from infective humans, the development of Plasmodium can only continue indoors, in heated buildings. Conclusion: Northern malaria existed in a cold climate by means of summer dormancy of hypnozoites in humans and indoor transmission of sporozoites throughout the winter by semiactive hibernating mosquitoes. Variable climatic conditions did not affect this relationship. The epidemics, however, were regulated by the population size of the mosquitoes which, in turn, ultimately was controlled by the temperatures of the preceding summer.