Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces

Maximal monotone operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a maximal monotone operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces was obtained. In this work we study some others conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces.