Harmonic Analysis Seminar (8,7)

Harmonic Analysis Seminar
Date and Time: 10:00am-12:00pm, August 7, 2015
Location: Room 1129, New Main Building
Speaker: Professor Yuan Xu, University of Oregon, USA
Title: Uncertainty Principle on the Unit Sphere and Related Domains
 Abstract: We discuss recent results on uncertainty principle on the unit sphere in the Euclidean space, which is an analogue of the classical Heisenberg inequality. We give two proofs, the first one is based on the Hardy-Rellich inequality on the sphere, which does not hold, however, for the unit sphere in R^3, the second one holds for all dimensions but give a worse constant when the dimension is large. We can also establish such uncertainty principles in the setting of weighted unit sphere for a family of weight functions invariant under a reflection group, and on weighted unit ball and simplex. 
The talk will contain two parts. The first part will be on unweighted inequalities on the unit sphere, based on jointed work with Feng Dai; the second part is on weighted inequalities on the sphere, and on inequalities on the unit ball and the standard simplex. Along the way, we shall introduce the basics of the theory of h-harmonics developed by Dunkl and the connection between orthogonal structures on the unit sphere, unit ball and the simplex.