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

13 Jan 2020 Nguyen Van Hoang

Let $2\leq m < n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ 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 $W^m L^{\frac nm,q}(\mathbb H^n)$ \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}\leq 1} \int_{\mathbb H^n} \Phi_{\frac nm,q}\big(\beta_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] 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... (read more)

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