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.

Periodic solutions of abstract neutral functional differential equations

We characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a UMD space. (C) 2011 Elsevier B.V. All rights reserved.

Periodic solutions of abstract neutral functional differential equations

We characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a UMD space. (C) 2011 Elsevier Inc. All rights reserved.

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.

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.

MiLK: Transforming everyday places into dynamic learning spaces with mobile and web technologies

This paper explores how game authoring tools can teach processes that transform everyday places into engaging learning spaces. It discusses the motivation inherent in playing games and creating games for others, and how this stimulates an iterative process of creation and reflection and evokes a natural desire to engage in learning. The use of MiLK at the Adelaide Botanic Gardens is offered as a case in point. MiLK is an authoring tool that allows students and teachers to create and share SMS games for mobile phones. A group of South Australian high school students used MiLK to play a game, create their own games and play each other’s games during a day at the gardens. This paper details the learning processes involved in these activities and how the students, without prompting, reflected on their learning, conducted peer assessment, and engaged in a two-way discussion with their teacher about new technologies and their implications for learning. The paper concludes with a discussion of the needs and requirements of 21st century learners and how MiLK can support constructivist and connectivist teaching methods that engage learners and will produce an appropriately skilled future workforce.

Designing learning spaces for (partially) online lives : recombinant architecture

The notion of recombinant architecture signals a loosening of spatial connections between physical and digital-online environments (Mitchell, 1996; 2000; 2003). Such an idea also points to the transformative nature of the designing approaches concerned with the creation of spaces where bits meet bodies to fulfil human needs and desires and, at the same time, pursuing those human dimensions of space and place which are so important to our senses of belonging, physical comfort and amenity. This paper proposes that recombinant spaces and places draw on familiar architectural forms and functions and on the transforming functions of digital-online modes. Perspectives, approaches and resources outlined in the paper support designing and re-designing enterprises and aim to stimulate discussion in the Digital Environments strand of this online conference: 'Under Construction: a world without walls'.

There is nothing that 'identifies me to that place' : Aboriginal Women's perceptions of health spaces and places

There is a growing body of literature within social and cultural geography that explores notions of place, space, culture, race and identity. The more recent works suggest that places are experienced and understood in multiple ways and are embedded within an array of politics. Memmott and Long, who have undertaken place-based research with Australian Indigenous people, present the theoretical position that ‘place is made and takes on meaning through an interaction process involving mutual accommodation between people and the environment’. They outline that places and their cultural meanings are generated through one or a combination of three types of people–environment interactions. These include: a place that is created by altering the physical characteristics of a piece of environment and which might encompass a feature or features which are natural or made; a place that is created totally through behaviour that is carried out within a specific area, therefore that specific behaviour becomes connected to that specific place; and a place created by people moving or being moved from one environment to another and establishing a new place where boundaries are created and activities carried out. All these ideas of places are challenged and confirmed by what Indigenous women have said about their particular use of, and relationship with, space within several health services in Rockhampton, Central Queensland. As my title suggests, Indigenous women do not see themselves as ‘neutral’ or ‘non-racialised’ citizens who enter and ‘use’ a supposedly neutral health service. Instead, Aboriginal women demonstrate they are active recognisers of places that would identify them within the particular health place. That is, they as Aboriginal women didn’t just ‘make’ place, the places and spaces ‘make’ them. The health services were identified as sites within which spatial relations could begin to grow with recognition of themselves as Aboriginal women in place, or instead create a sense of marginality in the failure of the spaces to identify them. The women’s voices within this paper are drawn from interviews undertaken with twenty Aboriginal women in Rockhampton, Central Queensland, Australia, who participated in a research project exploring ‘how the relationship between health services and Aboriginal women can be more empowering from the viewpoints of Aboriginal women’. The assumption underpinning this study was that empowering and re-empowering practices for Aboriginal women can lead to improved health outcomes. Throughout the interviews women shared some of their lived realities including some of their thoughts on identity, the body, employment in the health sector, service delivery and their notions of health service spaces and places. Their thoughts on health service spaces and places provide an understanding of the lived reality for Aboriginal women and are explored and incorporated within this paper.

Recent Italian cinema : spaces, contexts, experiences

The book is an in-depth view of recent Italian cinema - bringing an interdisciplinary knowledge to the study of a complex cinema industry. The book aims to address a number of questions about Italian cinema of the last twenty years, bringing interdisciplinary knowledge to a cinema that eschews traditional definitions and categories, and challenges critical assumption about a film industry that is struggling to find a new direction. In doing so, Recent Italian Cinema offers a transverse analysis of the Italian cinema industry in its dealings with national and international production, and of the themes and issues that have emerged in films produced during the period 1980-2006.

Spaces of care in the third sector : Understanding the effects of professionalization

Increasingly the health and welfare needs of individuals and communities are being met by third sector, or not-for-profit, organizations. Since the 1980s third sector organizations have been subject to significant, sector-wide changes, such as the development of contractual funding and an increasing need to collaborate with governments and other sectors. In particular, the processes of ‘professionalization’ and ‘bureaucratization’ have received significant attention and are now well documented in third sector literature. These processes are often understood to create barriers between organizations and their community groups and neutralize alternative forms of service provision. In this article we provide a case study of an Australian third sector organization undergoing professionalization. The case study draws on ethnographic and qualitative interviews with staff and volunteers at a health-based third sector organization involved in service provision to marginalized community groups. We examine how professionalization alters organizational spaces and dynamics and conclude that professionalized third sector spaces may still be ‘community’ spaces where individuals may give and receive care and services. Moreover, we suggest that these community spaces hold potential for resisting the neutralizing effects of contracting.

Bringing the internet down to earth : emerging spaces of locative media

Web 1.0 referred to the early, read-only internet; Web 2.0 refers to the ‘read-write web’ in which users actively contribute to as well as consume online content; Web 3.0 is now being used to refer to the convergence of mobile and Web 2.0 technologies and applications. One of the most important developments in mobile 3.0 is geography: with many mobile phones now equipped with GPS, mobiles promise to “bring the internet down to earth” through geographically-aware, or locative media. The internet was earlier heralded as “the death of geography” with predictions that with anyone able to access information from anywhere, geography would no longer matter. But mobiles are disproving this. GPS allows the location of the user to be pinpointed, and the mobile internet allows the user to access locally-relevant information, or to upload content which is geotagged to the specific location. It also allows locally-specific content to be sent to the user when the user enters a specific space. Location-based services are one of the fastest-growing segments of the mobile internet market: the 2008 AIMIA report indicates that user access of local maps increased by 347% over the previous 12 months, and restaurant guides/reviews increased by 174%. The central tenet of cultural geography is that places are culturally-constructed, comprised of the physical space itself, culturally-inflected perceptions of that space, and people’s experiences of the space (LeFebvre 1991). This paper takes a cultural geographical approach to locative media, anatomising the various spaces which have emerged through locative media, or “the geoweb” (Lake 2004). The geoweb is such a new concept that to date, critical discourse has treated it as a somewhat homogenous spatial formation. In order to counter this, and in order to demonstrate the dynamic complexity of the emerging spaces of the geoweb, the paper provides a topography of different types of locative media space: including the personal/aesthetic in which individual users geotag specific physical sites with their own content and meanings; the commercial, like the billboards which speak to individuals as they pass in Minority Report; and the social, in which one’s location is defined by the proximity of friends rather than by geography.