Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

Abstract

We establish sharp Hardy–Adams inequalities on hyperbolic space B4 of dimension four. Namely, we will show that for any α>0 there exists a constant Cα>0 such that∫B4(e32π2u2−1−32π2u2)dV=16∫B4e32π2u2−1−32π2u2(1−|x|2)4dx≤Cα for any u∈C0∞(B4) with∫B4(−ΔH−94)(−ΔH+α)u⋅udV≤1. As applications, we obtain a sharpened Adams inequality on hyperbolic space B4 and an inequality which improves the classical Adams' inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy–Trudinger–Moser inequality on a disk in dimension two given by Wang and Ye in [46] and on any convex planar domain by Lu and Yang in [33].The Fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work.