Comparison of Two Weak Versions of the Orlicz Spaces

In this work two versions of weak Orlicz spaces that appear in the literature, MA and MA, are analyzed. One of those include the weak Lebesgue spaces for 1 :5 p < 00, while the other version gives these spaces only for p > 1, resulting the stronger space L1 in the extrem case p = 1. Necessary and sufficient condit.ions about t.he growth function A in order that. both spaces coincide arc given. Moreover we prove that these same conditions characterize the normability of the MA space. . I.INTRODUCTION. We shall denote by MA the weak Orlicz space associated to A, defined as in the work of O'Neil, [0], where he makes use of this kind of functions to obtain a generalization of the Hardy-Littlewood-Sobolev's theorem on fractional integration into the context of Orlicz spaces. This version of weak Orlicz spaces generalizes the weak LP spaces, L~, but only for p > 1. In fact the class MA for A the identity function gives a proper subspace of L!. Our aim in t.his work is to present an alternative definition of a weak Orlicz space associated to the function A, denoted by MA, in order to include allL~ for 1 ::; p < 00. In this way our spaces MA give L! for A the identity function and they coincide with MA for A(t) = t P , p > 1. Moreover we shall prove that both spaces are exactly the same as long as A keeps a "little bit away" from the identity. In fact we establish in theorem (4.8) the necessary and sufficient conditions on A to guarantee the equality MA = MA. We would like to point out that the spaees MA are easier to handle since they are defined in terms of a norm while in turn, MA is given by means of a quantity which is not necessarily a norm. It is well known that the weak LP spaces are Supported by: CONICET and UNL (CAI+D Program).