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 A(2)(D), namely f is an element of L-2(D), if and only if parallel to f parallel to L-1+ H-1/2(S-1) := limsup(r -> 1)parallel to f(re(i theta))parallel to L-1+ H-1/2(S-1) < +infinity. Possible generalisations to the higher-dimensional torus are explored. (C) 2021 The Authors. Published by Elsevier Inc.
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