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.

The effect of adding features on product attractiveness: the role of product perceived congruity

Technological progress has been enabling companies to add disparate features to their existing products. This research investigates the effect of adding more features on consumers’ evaluation of the product, by examining in particular the role of the congruity of the features added with the base product as a variable the moderates the effect of increasing the number of features. Grounding on schema-congruity theory, I propose that the cognitive elaboration associated with the product congruity of the features added explains consumers’ evaluation as the number of new features increases. In particular, it is shown that consumers perceive a benefit from increasing the number of features only when these features are congruent with the product. The underlying mechanisms that explains this finding predicts that when the number of incongruent features increases the cognitive resources necessary to elaborate such incongruities increase and consumers are not willing to spend such resources. However, I further show that when encouraged to consider the new features thoughtfully, consumers do seem able to infer value from increasing the number of moderately incongruent features. Nonetheless, this finding does not apply for those new features that are extremely incongruent with the product. Further evidence for consumers’ ability to resolve the moderate incongruity associated with adding more features is also shown, by studying the moderating role of temporal construal. I propose that consumers perceive an increase in product evaluation as the number of moderately incongruent features increases when consumers consider purchasing the product in the distant future, whereas such an increase is not predicted for the near future scenario. I verify these effect in three experimental studies. Theoretical and managerial implications, and possible avenues of future research are also suggested.