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## Annales Polonici Mathematici

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## Singular Hardy–Adams inequalities on hyperbolic spaces of any even dimension

### Volume 129 / 2022

Annales Polonici Mathematici 129 (2022), 175-192 MSC: Primary 46E35; Secondary 35J20, 42B35. DOI: 10.4064/ap211020-29-4 Published online: 24 August 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.

#### Authors

• Xumin WangCollege of Science
Beijing Forestry University
Beijing 100083, P.R. China
e-mail
• Guanglan WangSchool of Mathematics and Statistics
Linyi University
Linyi 276005, P.R. China
e-mail

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