Lemma 6.6 (Liouville theorem on ℝ m + n ) . [0553] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Lemma 6.6 (Liouville theorem on ℝ m + n \mathbb{R}^{m+n} .
Given m , n ∈ ℤ + m,n\in\mathbb{Z}_{+} m + n ≥ 3 m+n\geq 3 μ p ∈ ( − 1 , 1 ) ∖ { 0 } \mu_{p}\in(-1,1)\setminus\{0\} u ∈ C ∞ ( ℝ m + n ) u\in C^{\infty}(\mathbb{R}^{m+n}) ( ℝ m + n , g ℝ m ⊕ g ℝ n ) (\mathbb{R}^{m+n},g_{\mathbb{R}^{m}}\oplus g_{\mathbb{R}^{n}}) u u
(6.42) | u ( x , y ) | ≤ C | x | μ p , ∀ ( x , y ) ∈ ( ℝ m ∖ { 0 } ) × ℝ n , \displaystyle|u(x,y)|\leq\frac{C}{|x|^{\mu_{p}}},\ \forall(x,y)\in(\mathbb{R}^{m}\setminus\{0\})\times\mathbb{R}^{n},
then u ≡ 0 u\equiv 0 ℝ m + n \mathbb{R}^{m+n}