Bergman-Bourgain-Brezis-type inequality

Abstract

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:D→C belongs to the Bergman space A2(D), namely f∈L2(D), if and only if‖f‖L1+H−1/2(S1):=limsupr→1−‖f(reiθ)‖L1+H−1/2(S1)<+∞. Possible generalisations to the higher-dimensional torus are explored.