Abstract For any non-zero function u in the Schwartz space S ( R n ) , we prove that s ↦ [ u ] s , 2 2 can be extended to C as a transcendental meromorphic function, which establishes a connection between the Bourgain-Brezis-Mironescu's formula, Maz'ya-Shaponshikova's formula and the residues of the transcendental meromorphic function of s ↦ [ u ] s , 2 2 at s = 0 and s = 1 separately. Moreover, we study the function properties of [ u ] s , 2 2 and obtain the convergence rate version of the Bourgain-Brezis-Mironescu's formula and Maz'ya-Shaponshikova's formula. And we also obtain the following sharp interpolation inequality. For any u ∈ W 1 , 2 ( R n ) and s ∈ ( 0 , 1 ) , we have [ u ] s , 2 2 ≤ π n 2 + 1 2 2 s − 1 Γ ( n 2 + s ) Γ ( 1 + s ) sin ⁡ π s ‖ u ‖ 2 2 ( 1 − s ) ‖ ∇ u ‖ 2 2 s .

中文翻译：

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从复杂分析的角度看分数 Sobolev 空间

摘要 对于 Schwartz 空间 S ( R n ) 中的任何非零函数 u，我们证明 s ↦ [ u ] s , 2 2 可以作为超越亚纯函数扩展到 C，从而建立 Bourgain-Brezis 之间的联系-Mironescu 公式、Maz'ya-Shaponshikova 公式和 s ↦ [ u ] s , 2 2 在 s = 0 和 s = 1 的超越亚纯函数的残数。此外，我们研究了[ u ] s , 2 2 的函数性质，得到了Bourgain-Brezis-Mironescu 公式和Maz'ya-Shaponshikova 公式的收敛速度版本。我们还得到以下尖锐的插值不等式。对于任何 u ∈ W 1 , 2 ( R n ) 和 s ∈ ( 0 , 1 ) ，我们有 [ u ] s , 2 2 ≤ π n 2 + 1 2 2 s − 1 Γ ( n 2 + s ) Γ ( 1 + s ) sin ⁡ π s ‖ u ‖ 2 2 ( 1 − s ) ‖ ∇ u ‖ 2 2 s 。