English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this article we prove Adams inequality with exact growth condition in thefour dimensional hyperbolic space $\mathbb{H}^4,$  \begin{align}  \int_{\mathbb{H}^4} \frac{e^{32 \pi^2 u^2} - 1}{(1 + |u|)^2} \ dv_g \leq C||u||^2_{L^2({\mathbb{H}^4})}.  \end{align} for all $u \in C^{\infty}_c(\mathbb{H}^4) $ with$\int_{\mathbb{H}^4} (P_2 u) u \ dv_g \leq 1.$  We will also establish an Adachi-Tanaka type inequality in this settings.Another aspect of this article is the P.L.Lions lemma in the hyperbolic space.We prove P.L.Lions lemma for the Moser functional and for a few cases of theAdams functional on the whole hyperbolic space.

Adams' inequality with exact growth in the hyperbolic space $\mathbb{H}^4$ and Lions lemma