Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev inequalities

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for Gagliardo-Nirenberg-Sobolev inequalities $\|w\r|_q \leq C\|\nabla w\|_2^\theta \|w\|_p^{1-\theta}$. These invariants are constructed via a minimization procedure for the weighted scalar curvature functional in the conformal class of a smooth metric measure space. We then describe critical points which are also critical points for variations in the metric or the measure. When the measure is assumed to take a special form --- for example, as the volume element of an Einstein metric --- we use this description to show that minimizers of our invariants are only critical for certain values of $p$ and $q$. In particular, on Euclidean space our result states that either $p=2(q-1)$ or $q=2(p-1)$, giving a new characterization of the GNS inequalities whose sharp constants were computed by Del Pino and Dolbeault.Comment: 20 page

Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian

We establish a family of sharp Sobolev trace inequalities involving the $W^{k,2}(\mathbb{R}_+^{n+1},y^a)$-norm. These inequalities are closely related to the realization of fractional powers of the Laplacian on $\mathbb{R}^n=\partial\mathbb{R}_+^{n+1}$ as generalized Dirichlet-to-Neumann operators associated to powers of the weighted Laplacian in upper half space, generalizing observations of Caffarelli--Silvestre and of Yang.Comment: 25 page

Delivering knowledge in the field: A telecommunications service provision and maintenance case

This paper proposes a novel approach to providing knowledge management services in a business process wherein field engineers are the main process actors, providing and maintaining telecommunications services. Cooperating multi-agents play a central role for the provision of knowledge management services by integrating heterogeneous systems to collect related knowledge for the execution of mobile tasks. The proposed system is expected to increase both the performance of the mobile workforce and customer satisfaction by supporting and encouraging knowledge sharing