Publication

Abstract : In this article we explore a new growth condition on Young functions, which we call Mulholland condition, pertaining to the mathematician H.P Mulholland, who studied these functions for the first time, albeit in a different context. We construct a non-trivial Young function Ω which satisfies Mulholland condition and Δ 2 -condition. We then associate exotic F -norms to the vector space X 1 ⊕ X 2 , where X 1 and X 2 are Banach spaces, using the function Ω . This F -spaces contains the Banach space X 1 and X 2 as a maximal Banach subspace. Further, the Banach envelope ( X 1 ⊕ X 2 , | | . | | Ω o ) of this F -space corresponds to the Young function Ω o who characteristic function is an asymptotic line to the characteristic function of the Young function Ω . Thus these F -spaces serves as "interpolation space" for Banach spaces X 1 and ( X 1 ⊕ X 2 , | | . | | Ω o ) in some sense. These F -space are well behaved in regards to Hahn-Banach extension property, which is lacking in classical F -spaces like L p and H p for 0 < p < 1 . Towards the end, some direct sums for Orlicz spaces are discussed.