Fourier analysis and optimal Hardy-Adams inequalities on hyperbolic spaces of any even dimension

The main purpose of this paper is to use a substantially new method of estimating the Hardy operator to establish the sharp Hardy-Adams inequalities on hyperbolic spaces Bn for all even dimension n and n≥4. As applications of such inequalities, we will improve substantially the known Adams inequalities on hyperbolic space Bn in the literature and also strengthen the classical Adams' inequality and the Hardy inequality on Euclidean balls in any even dimension. The later inequality can be viewed as the borderline case of the sharp Hardy-Sobolev-Maz'ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors [41].