On some Fr\'echet spaces associated to the functions satisfying Mulholland inequality

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 $\Omega$ which satisfies Mulholland condition and $\Delta_2$-condition. We then associate exotic $F$-norms to the vector space $X_1\oplus X_2$, where $X_1$ and $X_2$ are Banach spaces, using the function $\Omega$. This $F$-spaces contains the Banach space $X_1$ and $X_2$ as a maximal Banach subspace. Further, the Banach envelope $(X_1\oplus X_2,||.||_{\Omega_o})$ of this $F$-space corresponds to the Young function $\Omega_o$ who characteristic function is an asymptotic line to the characteristic function of the Young function $\Omega$. Thus these $F$-spaces serves as "interpolation space" for Banach spaces $X_1$ and $(X_1\oplus X_2, ||.||_{\Omega_o})$ 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.