The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms

Let $2\le m<n$ and $q\in (1,\mathrm{\infty })$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$theLorentz-Sobolevspaceoforder$m$ in the hyperbolic space ${\mathbb{H}}^n$. In this paper, we establish the following Adams inequality in the Lorentz-Sobolev space $W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n)$ $$\underset{u\in W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n),\mathrm{\parallel }{\mathrm{\nabla }}_g^{m}u{\mathrm{\parallel }}_{\frac{n}{m},q}\le 1}{\sup }{\int }_{{\mathbb{H}}^n}{\mathrm{\Phi }}_{\frac{n}{m},q}\left({\beta }_{n,m}^{\frac{q}{q-1}}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{q}{q-1}}\right)d{V}_g<\mathrm{\infty }$$ for $q\in (1,\mathrm{\infty })$ if $m$ is even, and $q \in (1,n/m)$if$m$isodd,where$\beta_{n,m}^{q/(q-1)}$ is the sharp exponent in the Adams inequality under Lorentz-Sobolev norm in the Euclidean space. To our knowledge, much less is known about the Adams inequality under the Lorentz-Sobolev norm in the hyperbolic spaces. We also prove an improved Adams inequality under the Lorentz-Sobolev norm provided that $q\ge 2n\mathrm{/}(n-1)$ if $m$ is even and $2n\mathrm{/}(n-1)\le q\le \frac{n}{m}$ if $m$ is odd, $$\underset{u\in W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n),\mathrm{\parallel }{\mathrm{\nabla }}_g^{m}u{\mathrm{\parallel }}_{\frac{n}{m},q}^q-\lambda \mathrm{\parallel }u{\mathrm{\parallel }}_{\frac{n}{m},q}^q\le 1}{\sup }{\int }_{{\mathbb{H}}^n}{\mathrm{\Phi }}_{\frac{n}{m},q}\left({\beta }_{n,m}^{\frac{q}{q-1}}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{q}{q-1}}\right)d{V}_g<\mathrm{\infty }$$ for any $0<\lambda <C(n,m,n\mathrm{/}m{)}^q$ where $C(n,m,n\mathrm{/}m{)}^q$ is the sharp constant in the Lorentz-Poincar'e inequality. Finally, we establish a Hardy-Adams inequality in the unit ball when $m\ge 3$, $n\ge 2m+1$ and $q\ge 2n\mathrm{/}(n-1)$ if $m$ is even and $2n\mathrm{/}(n-1)\le q\le n\mathrm{/}m$ if $m$ is odd $$\underset{u\in W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n),\mathrm{\parallel }{\mathrm{\nabla }}_g^{m}u{\mathrm{\parallel }}_{\frac{n}{m},q}^q-C(n,m,\frac{n}{m}{)}^q\mathrm{\parallel }u{\mathrm{\parallel }}_{\frac{n}{m},q}^q\le 1}{\sup }{\int }_{{\mathbb{B}}^n}\exp \left({\beta }_{n,m}^{\frac{q}{q-1}}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{q}{q-1}}\right)dx<\mathrm{\infty }\mathrm{.}$$