Event № 247
Scaling transformations (translations and dilations) are known to define wavelet bases, give equivalent definitions of important functional spaces, and prove optimal inequalities. We will summarize some known results in the Euclidean case and on nilpotent Lie groups, and discuss the work in progress dealing with analogous transformations on manifolds, where scaling is defined via the Green's function of Laplace-Beltrami operator. Preliminary results include sharp inequalities of Caffarelli-Kohn-Nirenberg type on the hyperbolic space. The work involves collaborations with L. Skrzypczak and K. Sandeep.