Suppose $\Phi(t)=\int_0^t \phi(s)ds$ is an Young function with $\phi \in C^1 (0,\infty)$. Assume also that ($\Delta_2$ condition)\begin{equation}\tag{1}(1+\Gamma_1)\Phi(t)\leq t \p …

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0, …