论文标题
尖锐的亚当人类型的不平等现象在Lorentz-Sobolev Norms下的双曲线空间中
The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms
论文作者
论文摘要
令$ 2 \ leq m <n $和$ q \ in（1，\ infty）$，我们用$ w^ml^{\ frac nm，q}（\ Mathbb h^n）$ lorentz-sobolev $ \ mathbb h^n $中的lorentz-sobolev $ m $。在本文中，我们在Lorentz-Sobolev空间中建立以下ADAM不等式u \ | _ {\ frac nm，q} \ leq 1} \ int _ {\ Mathbb h^n}φ_{\ frac nm，q} \ big（β_{N，m} \infty \] for $q \in (1,\infty)$ if $m$ is even, and $q \in (1,n/m)$ if $m$ is odd, where $β_{n,m}^{q/(q-1)}$ is the sharp exponent in the Adams inequality under Lorentz-Sobolev norm in the Euclidean space.据我们所知，对双曲线空间中Lorentz-Sobolev Norm下的Adams不平等的知之甚少。我们还证明了在Lorentz-Sobolev Norm下的ADAMS不平等，但如果$ Q \ Q \ geq 2n/（n-1）$如果$ m $均匀，$ 2N/（N-1）\ leq Q \ leq Q \ leq Q \ leq \ frac nm $如果奇数是奇数，\ [\ sup_ {\ sup_ { h^n），\，\ | \ nabla_g^m u \ \ | _ {{\ frac nm，q}^q -λ\ | u \ | _ \ | _ {\ frac nm，q}^Q}^q \ leq 1} nm，q} \ big（β_{n，m}^{\ frac q {q-1}} | U |^{\ frac q {q-1}} \ big） Lorentz-Poincaré不平等。最后，当$ m \ geq 3 $，$ n \ geq 2m+1 $+1 $和$ q \ geq 2n/（n-1）$如果$ m $均匀且$ 2N/（n-1）\ leq q \ leq q \ leq n/m $如果$ m $ odd odd odd odd odd odd odd odd odd odd odd odd odd odd odd odd $ f \ f iq f iq fore nm，q}（\ Mathbb h^n），\，\ | \ nabla_g^m u \ \ | _ {\ frac nm，q}^q -c（n，m，m，m，\ frac nm）^q \ | U \ | \ exp \ big（β_{n，m}^{\ frac q {q-1}} | u |^{\ frac q {q-1}} \ big）dx <\ infty。 \]
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 $W^m L^{\frac nm,q}(\mathbb H^n)$ \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}\leq 1} \int_{\mathbb H^n} Φ_{\frac nm,q}\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] for $q \in (1,\infty)$ if $m$ is even, and $q \in (1,n/m)$ if $m$ is odd, where $β_{n,m}^{q/(q-1)}$ 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 $q\geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq \frac nm$ 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 $0< λ< C(n,m,n/m)^q$ 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 $m\geq 3$, $n\geq 2m+1$ and $q \geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq n/m$ if $m$ is odd \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb B^n} \exp\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dx < \infty. \]
