Singular Hardy–Adams inequalities on hyperbolic spaces of any even dimension
Volume 129 / 2022
Abstract
The main purpose of this paper is to establish singular Hardy–Adams inequalities on hyperbolic space $\mathbb B^n$ of any even dimension. More precisely, we will prove that for any even integer $n\geq 4$, $\alpha \gt 0$ and $0 \lt \beta \lt n$, there exists a constant $C_{\alpha ,\beta } \gt 0$ such that for $u \in C_0^\infty (\mathbb B^n)$ satisfying $$ \int _{\mathbb B^n}\bigl (-\Delta _{\mathbb H}-(n-1)^2/4\big ) \big (-\Delta _{\mathbb H}-(n-1)^2/4+\alpha ^2\big )^{n/2-1}u\cdot u\, dV\leq 1, $$ we have $$ \int _{\mathbb B^n} \frac {\exp \bigl (\beta _0(n/2,n)(1-\beta /n)u^2\big )-1-\beta _0(n/2,n)(1-\beta /n)u^2}{\rho (x)^\beta }\,dV\leq C_{\alpha ,\beta }, $$ where $\rho (x)=\log \frac {1+|x|}{1-|x|}$. Our methods are based on the Helgason–Fourier analysis on real hyperbolic spaces and Green function estimates for the associated differential operators developed in a series of works by Li, Lu and Yang. However, the presence of the singular term $\frac {1}{\rho (x)^\beta }$ adds some additional work and difficulty. Here, we apply the properties of the density function $J(\theta ,\rho )$ to overcome the presence of the weight $\frac {1}{\rho (x)^\beta }$. As an application, we also prove some sharpened Adams inequalities on $\mathbb B^n$. These results extend that of Li, Lu and Yang [Adv. Math. 333 (2018), 350–385] to the weighted situation.
