Mathematics

Dr. Debdip Ganguly
Technion - Israel Institute of Technology, Haifa

In this talk, Hardy-type inequalities associated to the quadratic form of the shifted Laplacian −∆HN − (N − 1)2/4 on the hyperbolic space HN , (N − 1)2/4 being, as it is well-known, the bottom of the L2 -spectrum of −∆HN will be presented. Sharpness of constants of the resulting Poincar´e-Hardy inequality and the criticality of the operator will also be discussed. Furthemore a related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, will be considered. It involves an explicit, curvature dependent and typically unbounded potential, and the resulting Schr¨odinger operator will be shown to be critical in a suitable sense. If time permits I will also consider higher order analogue of improved Poincar´e inequality and discuss related open questions.