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

Abstract

Let $2\le m<n$ and $q\in (1,\infty )$, we denote by $W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n)$ the Lorentz-Sobolev space of order $m$ in the hyperbolic space ${\mathbb{H}}^n$. In this paper, we establish the following Adams inequality in the Lorentz-Sobolev space $W^m{L}^{\frac{n}{m},q}({\mathbb{H}}^n)$ \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}\leq 1} \int_{\mathbb H^n} Φ_{\frac nm,q}\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] for $q\in (1,\infty )$ if $m$ is even, and $q\in (1,n/m)$ if $m$ is odd, 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 $q\ge 2n/(n-1)$ if $m$ is even and $2n/(n-1)\le q\le \frac{n}{m}$ if $m$ is odd, \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}^q -λ\|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb H^n} Φ_{\frac nm,q}\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] for any $0<\lambda <C(n,m,n/m{)}^q$ where $C(n,m,n/m{)}^q$ is the sharp constant in the Lorentz-Poincaré inequality. Finally, we establish a Hardy-Adams inequality in the unit ball when $m\ge 3$, $n\ge 2m+1$ and $q\ge 2n/(n-1)$ if $m$ is even and $2n/(n-1)\le q\le n/m$ if $m$ is odd \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb B^n} \exp\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dx < \infty. \]

27 pages, comment are welcomearXiv admin note: text overlap with arXiv:2001.03950

References in corpus (1)

• A sharp Adams inequality in dimension four and its extremal functions