Bergman-Bourgain-Brezis-type Inequality

Abstract

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 $$\Vert f{\Vert }_{L^1+H^{-1/2}(S^1)}:=\underset{r\to 1^{-}}{\mathrm{limsup}}\Vert f(r{e}^{i\theta }){\Vert }_{L^1+H^{-1/2}(S^1)}<+\infty .$$ Possible generalisations to the higher-dimensional torus are explored.