标题： 关于满足Mulholland不等式的函数相关的某些Fréchet空间 显示英文标题

标题： On some Fréchet spaces associated to the functions satisfying Mulholland inequality

摘要： 在本文中，我们探讨了一种关于Young函数的新增长条件，我们称之为Mulholland条件，这是以数学家H.P Mulholland命名的，他首次研究了这些函数，尽管是在不同的背景下。 我们构造了一个非平凡的Young函数$\Omega$，它满足Mulholland条件和$\Delta_2$-条件。 然后，我们将奇特的$F$-范数与向量空间$X_1\oplus X_2$相关联，其中$X_1$和$X_2$是巴拿赫空间，使用函数$\Omega$。 此$F$-空间包含巴拿赫空间$X_1$和$X_2$作为极大巴拿赫子空间。 进一步，此$F$-空间的巴拿赫包络$(X_1\oplus X_2,||.||_{\Omega_o})$对应于 Young 函数$\Omega_o$，其特征函数是 Young 函数$\Omega$特征函数的渐近线。 因此这些$F$空间在某种意义上作为巴拿赫空间$X_1$和$(X_1\oplus X_2, ||.||_{\Omega_o})$的“插值空间”。 这些$F$-空间在Hahn-Banach扩展性质方面表现良好，这在经典的$F$-空间如$L^p$和$H^p$中是缺乏的，对于$0<p<1$来说也是如此。最后，讨论了一些Orlicz空间的直接和。 显示英文摘要

摘要： 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.