Let $$\mathbb{B}$$ be the unit disc in ℝ2, ℋ be the completion of $$C_0^\infty \left( \mathbb{B} \right)$$ under the norm $$\parallel{u}\parallel_\mathscr{H}=(\int_{\mathbb{B}}|\triangledown{u}|^2dx-\int_{\mathbb{B}}\frac{u^2}{(1-|x|^2)^2}dx)^\frac{1}{2}, \;\;\;\forall{u}\in{C_0^\infty}(\mathbb{B}).$$ By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric $${dv_{\mathscr{H}}} = {{dx} \over {{{\left( {1 - {{\left| x \right|}^2}} \right)}^2}}}$$, we prove that, for any fixed $$\alpha ,\,0 \le \alpha 1.$$ This is an analog of early results of Lu—Yang (Discrete Contin. Dyn. Syst., 2009) and Yang (Trans. Amer. Math. Soc., 2007), and extends those of Wang—Ye (Adv. Math., 2012) and Yang—Zhu (Ann. Global Anal. Geom., 2016).

中文翻译：

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双曲空间中涉及 Lp 范数的 Hardy-Trudinger-Moser 不等式

令$$\mathbb{B}$$为ℝ2中的单位圆盘，ℋ为$$C_0^\infty \left( \mathbb{B} \right)$$在范数$$\parallel{u下的完成}\parallel_\mathscr{H}=(\int_{\mathbb{B}}|\triangledown{u}|^2dx-\int_{\mathbb{B}}\frac{u^2}{(1-| x|^2)^2}dx)^\frac{1}{2}, \;\;\;\forall{u}\in{C_0^\infty}(\mathbb{B}).$$膨胀分析的方法和关于标准双曲度量的重排论证 $${dv_{\mathscr{H}}} = {{dx} \over {{{\left( {1 - {{\ left| x \right|}^2}} \right)}^2}}}$$，我们证明，对于任意固定的 $$\alpha ,\,0 \le \alpha 1.$$ 这是一个模拟Lu-Yang (Discrete Contin. Dyn. Syst., 2009) 和 Yang (Trans. Amer. Math. Soc., 2007) 的早期结果，并扩展了 Wang-Ye (Adv. Math., 2012) 和 Yang ——朱（Ann. Global Anal. Geom., 2016）。