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.

Suomussalmen varhaisproterotsooisten diabaasien mineralogia tektonisessa tulkinnassa

The object of this research is to study the mineralogy of the diabase dykes in Suomussalmi and the relevance of the mineralogy to tectonic events, specifically large block movements in the Archaean crust. Sharp tectonic lines separate two anomalies in the dyke swarms, shown on a geomagnetic map as positive anomalies. In one of these areas, the Toravaara anomaly, the diabases seem to contain pyroxenes as a main component. Outside the Toravaara anomaly hornblende is the main ferromagnesian mineral in diabases. The aim of this paper is to research the differences in the diabases inside and outside the anomalies and interpret the processes that formed the anomalies. The data for this sudy consist of field observations, 120 thin sections, 334 electron microprobe analyses, 19 whole-rock chemical analyses, a U-Pb age analysis and geomagnetic low-altitude aerial survey maps. The methods are interpretation of field observations, chemical analyses, microprobe analyses of single minerals and radiometric age determination, microscopic studies of the thin sections, geothermometers and geobarometers. On the basis of field observations and petrographic studies the diabases in the area are divided into pyroxene diabases, hornblende diabases and the Lohisärkkä porphyritic dyke swarm. Hornblende diabases are found in the entire study area, while the pyroxene diabases concentrate on the area of the Toravaara geomagnetic anomaly. The Lohisärkkä swarm transects the whole area as a thin line from east to west. The diabases are fairly homogenous both chemically and by mineral composition. The few exceptions are part of rarer older swarms or are significantly altered. The Lohisärkkä dyke swarm was dated as 2,21 Ga old, significantly older than the most common 1,98 Ga swarm in the area. The geothermometers applied showed that the diabases on the Toravaara anomaly were stabilized at a much higher temperature than the dykes outside the anomaly. The geobarometers showed the pyroxenes to have crystallized at varying depths. The research showed the Toravaara anomaly to have formed by a vertical block movement, and the fault on its west side to have a total lateral transfer of only a few kilometers. The formation of the second anomaly was also interpreted to be tectonic in nature. In addition, the results of the geothermobarometry uncovered necessary conditions for the study of diabase emplacement depth: the minerals for the study must be chosen by minimum crystallization depth, and a geobarometer capable of determining the magmatic temperature must be used. In addition, it would be more suitable to conduct this kind of study in an area where the dykes are more exposed.