In this note, we prove a fractional version in 1-D of the Bourgain-Brezis inequality [1]. We show that such an inequality is equivalent to the fact that a holomorphic function $f:\mathbb{D}\to \mathbb{C}$ belongs to the Bergman space ${\mathcal{A}}^2(\mathbb{D})$, namely $f\in L^2(\mathbb{D})$, if and only if${\Vert f\Vert }_{L^1+H^{-1/2}(S^1)}:=\underset{r\to 1^{-}}{\lim \sup }{\Vert f(r{e}^{i\theta })\Vert }_{L^1+H^{-1/2}(S^1)}<+\infty .$ Possible generalisations to the higher-dimensional torus are explored.

中文翻译：

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Bergman-Bourgain-Brezis型不等式

在本笔记中，我们证明了 Bourgain-Brezis 不等式 [1] 的一维分数形式。我们证明了这样的不等式等价于一个全纯函数$F：\mathbb{D}\to \mathbb{C}$ 属于伯格曼空间 ${\mathcal{一种}}^2(\mathbb{D})$，即 $F\in {升}^2(\mathbb{D})$，当且仅当${\Vert F\Vert }_{{升}^1+H^{-1/2}({秒}^1)}：=\underset{r\to 1^{-}}{\mathrm{林}\mathrm{超}}{\Vert F(r{\mathrm{电子}}^{\mathrm{一世}\theta })\Vert }_{{升}^1+H^{-1/2}({秒}^1)}<+\infty .$ 探讨了对高维环面的可能推广。