Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Middleware principles and design for the integration of ubiquitos mobile services

Nowadays, in Ubiquitous computing scenarios users more and more require to exploit online contents and services by means of any device at hand, no matter their physical location, and by personalizing and tailoring content and service access to their own requirements. The coordinated provisioning of content tailored to user context and preferences, and the support for mobile multimodal and multichannel interactions are of paramount importance in providing users with a truly effective Ubiquitous support. However, so far the intrinsic heterogeneity and the lack of an integrated approach led to several either too vertical, or practically unusable proposals, thus resulting in poor and non-versatile support platforms for Ubiquitous computing. This work investigates and promotes design principles to help cope with these ever-changing and inherently dynamic scenarios. By following the outlined principles, we have designed and implemented a middleware support platform to support the provisioning of Ubiquitous mobile services and contents. To prove the viability of our approach, we have realized and stressed on top of our support platform a number of different, extremely complex and heterogeneous content and service provisioning scenarios. The encouraging results obtained are pushing our research work further, in order to provide a dynamic platform that is able to not only dynamically support novel Ubiquitous applicative scenarios by tailoring extremely diverse services and contents to heterogeneous user needs, but is also able to reconfigure and adapt itself in order to provide a truly optimized and tailored support for Ubiquitous service provisioning.