论文标题
Bergman-Bourgain-Brezis型不平等
Bergman-Bourgain-Brezis-type Inequality
论文作者
论文摘要
在本说明中，我们证明了Bourgain-Brezis不平等的$ 1 $ d的分数版本\ cite {Bourgain1}。我们表明，这种不等式等同于以下事实：holomorthic函数$ f \ colon \ d \ to \ c $属于伯格曼空间$ {\ nathcal {\ nathcal {a}}}^2（\ d）$ {h}^{ - 1/2}（s^1）}：= \ limsup_ {r \ to 1^ - } \ | f（re^{iθ}）\ | _ {l^1+ {h}+ {h}^{ - 1/2}（s^1）}（s^1）} <+ \ infty por tor torus。
In this note, we prove a fractional version in $1$-D of the Bourgain-Brezis inequality \cite{bourgain1}. We show that such an inequality is equivalent to the fact that a holomorphic function $f\colon\D\to\C$ belongs to the Bergman space ${\mathcal{A}}^2(\D)$, namely $f\in L^2(\D)$, if and only if $$\|f\|_{ L^1+ {H}^{-1/2}(S^1)}:=\limsup_{r\to 1^-}\|f(re^{iθ})\|_{ L^1+ {H}^{-1/2}(S^1)}<+\infty.$$ Possible generalisations to the higher-dimensional torus are explored.
