Title: Fourier analysis on hyperbolic spaces and applications to sharp geometric inequalities

 

Abstract: We will use the techniques of Fourier analysis on hyperbolic spaces to establish optimal geometric inequalities which will be sharper than those known in the Euclidean spaces in the literature. These include the sharp Hardy-Adams inequalities on balls and higher Hardy-Sobolev-Mazya inequalities on upper half spaces or hyperbolic balls. The relationship between the best constants for the sharp Sobolev inequalities and the Hardy-Sobolev-Mazya inequalities will also be discussed.