An equivalent norm for Orlicz spaces is the Luxemburg norm, defined on $L\Phi$ by $$ \|f\|'_{\Phi }=\inf \left\{k\in (0,\infty )\left| \int _{X}\Phi \left(\frac{|f|}{k}\right)\,\mathrm{d}\mu\right.\leq 1\right\} $$ as it can be seen in the relevant Wikipedia page. Moreover, in the particular case of Sobolev Spaces, the norm can be given the structure of an integral. Recall that more precisely, Orlicz spaces generalize $L^p$ spaces (for $1<p<\infty)$ in the sense that if $ \Phi (t)=t^{p}$ the above norm reduces to the one for these spaces given by $\|f\|_{L^p(\Omega)}:= \left(\int_\Omega |f|^p \right)^{1/p}dx$
I am wondering if there is an equivalent Orlicz norm given by an integral so that we do not need use the concept of infimum.
Precisely I'd prefer a norm given, for example, by the following expression as it is more manageable $$ \|f\|_{\Phi }=\sup \left\{\|fg\|_{1}\left| \int \Psi \circ |g|\,\mathrm{d}\mu \right.\leq 1\right\}. $$ My question is motivated by the fact that by Lemma 2.3 in the paper "A minimum problem with free boundary in Orlicz spaces" by Sandra Martínez and Noemi Wolanski we have $$ \|u\|_{G} \le C \max \left\{\left(\int_{\Omega} G(|u|) \mathrm{d}x\right)^{\frac{1}{g_0+1}}, \left(\int_{\Omega} G(|u|) \mathrm{d}x\right)^{\frac{1}{g_1+1}} \right\}. $$ So, I was wondering if in the case that that $$ 0 < g_0 \le \frac{t \cdot g'(t)}{g(t)} \le g_1 \quad \forall \, t >0. $$ the right hand side can be thought as a somewhat equivalent norm.
Yes, there is an equivalent Orlicz norm exactly with the structure that you required, with the function $\Psi$ being
$$\Phi^*(t):=\sup_{s \in \mathbb{R}} \{st-\Phi(s)\}$$
i.e. the convex coniugate of $\Phi$. Indeed, if we set $$\|f\|_\Phi:=\sup \{\|fg\|_1: \int \Phi^*(|g|) d\mu \leq 1\}$$ then $\|f\|’_\Phi \leq \|f\|_\Phi \leq 2\|f\|’_\Phi$. For the proof see this note (the second inequality follows from a variant of Holder’s inequality, while the first one is more delicate).