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.

Interruption and Uncooperativeness in Academic ELF Group Work : An Application of Linear Unit Grammar

This study addresses the challenge of analyzing interruption in spoken interaction. It begins with my observation of eight hours of academic group work among speakers of English as a lingua franca (ELF) in a university course. Unlike the common findings of ELF research which underscore the cooperative orientation of ELF users, this particular group gave strong impressions of interruption and uncooperativeness as they prepared a scientific group presentation. In the effort to investigate these impressions, I found that no satisfactory method exists for systematically identifying and analyzing interruptions. A useful tool was found in Linear Unit Grammar or LUG (Sinclair & Mauranen 2006), which analyzes spoken interaction prospectively as linear text. In the course of transcribing one of the early group work meetings, I developed a model of LUG-based criteria for identifying individual instances of interruption. With this system in place, I was then able to evaluate the aggregate occurrences of interruption in the group work and identify co-occurring interactive features which further influenced the perception of uncooperativeness. Finally, these aggregate statistics directed a return to the data and a contextually sensitive, qualitative analysis. This research cycle illuminates the interactive features which contributed to my own impressions of uncooperativeness, as well as the group members orientations to their own interruptive practice.

Metsikön rakenteen ja hakkuiden vaikutus pintakasvillisuuteen Etelä-Suomen yleisimmillä kasvupaikkatyypeillä

Taman tutkielman tarkoituksena oli selvittaa metsikon rakenteen seka hakkuiden vaikutuksia pintakasvillisuuden lajikoostumukseen ja biomassaan Etela-Suomen lehtomaisilla, tuoreilla ja kuivahkoilla kankailla. Aineistona tassa tyossa on 8. valtakunnan metsien inventoinnin yhteydessa vuosina 1985–86 metsaluonnon ja ympariston tilan seurantaa varten perustetuista noin 3 000 pysyvasta koealasta poimittu otos. Pintakasvillisuuden lajisto muuttuu metsikon kehitysvaiheen mukaan. Hakkuu on huomattava hairio, joka aiheuttaa nopeita ja suuria muutoksia pintakasvillisuudessa. Pintakasvillisuutta on tarkasteltu lahinna lajiryhmittain (heinat, ruohot, varvut, sammalet seka jakalat). Kunkin lajiryhman peittavyyden eroavaisuuksia testattiin varianssianalyysilla kun selittavana muuttujana ovat luokittain metsikon ika ja edellisesta hakkuusta kulunut aika. Lajikohtaisia tarkasteluja on sen sijaan tehty kasvillisuuden ordinaatioanalyyseilla. Tassa kaytetty ordinaatiomenetelma on epametrinen moniulotteinen skaalaus (Non-metric multidimensional scaling, NMDS), jonka avulla voidaan tehda paatelmia kasvillisuuden rakenteen ekologisesta vaihtelusta ymparistomuuttujien suhteen. Harvennus- ja avohakkuiden vaikutuksia pintakasvillisuuteen myos mallinnettiin lajiryhmittain kayttaen yleistettyja lineaarisia malleja (Generalized linear models). Lajiryhmien peittavyyksien kehitysta mallinnettiin puuston pohjapinta-alan funktiona. Metsikon ian kasvaessa heinien ja ruohojen osuus pienenee, kun taas varpujen ja sammalten osuus lisaantyy. Harvennushakkuiden vaikutukset ovat lievempia kuin avohakkuiden eivatka ne useimmiten aiheuttaneet tilastollisesti merkittavia muutoksia pintakasvillisuuden peittavyyksissa. Avohakkuu sen sijaan on voimakkaampi ja aiheuttaa merkittavia muutoksia. Heinia ja ruohoja esiintyy hakkuun jalkeen enemman ja vastaavasti sammalet ja varvut taantuvat. Kasvillisuuden kokonaispeittavyys ja biomassa ovat suurimmillaan hakkaamattomissa metsikoissa. Harvennushakkuun jalkeen peittavyys ja biomassa voi kuitenkin hetkellisesti olla suurimmillaan kun harvennuksesta on kulunut muutama vuosi. Yleistetyt lineaariset mallit kuvasivat pintakasvillisuuden kehitysta metsikon pohjapinta-alan funktiona luotettavasti. Malleja voidaan kayttaa myos ennustamaan miten pintakasvillisuus kehittyy avohakkuun jalkeen. Malleja voidaan soveltaa esimerkiksi laskettaessa pintakasvillisuuden sitoman hiilen maaraa eriikaisissa metsissa. Niiden avulla voidaan myos arvioida esimerkiksi avohakkuuta voimaperaisemman energiapuun korjuun vaikutuksia pintakasvillisuuden runsauteen.