A pr 2 00 7 Spaces between H 1 and L 1 Wael Abu - Shammala and

Because of their similarities, but mainly because of their differences, it is a matter of interest to determine the relationship between the Hardy space H(R) and the space of integrable functions L(R). The purpose of this paper is to address some unanswered questions concerning the family of spaces Xs(R ) that lie between H and L, and thus gain a better understanding of the gap that separates them. The spaces Xs were introduced by Sweezy, see [9]. They form a nested family that starts at H = X1 and approaches L 1 0, the subspace of L 1 functions with vanishing integral, as s → ∞. Here we consider the whole range of Xs spaces. First, Xs = H 1 for 0 < s ≤ 1; also, X∞ = L 1 0, see [1]. Further, we show that, for f ∈ Xs,