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

Abstract

Recently, the second and third authors established the sharp Hardy–Adams inequalities of second order derivatives with the best constants on the hyperbolic space B4 of dimension four ([40], Adv. Math. (2017)). A key method used in dimension four in [40] is that the following Hardy operator when n=4 and k=2∫Bn|∇ku|2dx−∏i=1k(2i−1)2∫Bnu2(1−|x|2)2kdx can be decomposed exactly as the product of fractional Laplacians. However, such a decomposition in dimension four is no longer possible in the case n>4 and 2≤k<n2 and the situation here is thus considerably more difficult and complicated. Therefore, it remains open as whether a sharp Hardy–Adams inequality still holds on higher dimensional hyperbolic space Bn for n>4.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].