It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and $L^{G_2}$ be Orlicz spaces. When do we have $L^{G_1} \subset L^{G_2}$?

It seems that this result holds only (maybe) if the Orlicz spaces $L^{G_1} \subset L^{G_2}$ are regular, that is, satisfy the famous $\Delta_2$ and $\nabla_2$ conditions. That is,
let's assume the conditions for N-functions introduced by G. Lieberman (see \cite{Lieberman}). More precisely,
$$
G'(t) = g(t), \ \ \text{with} \ \ g \in C^{0}([0,+ \infty]) \cap C^{1}((0,+ \infty])
$$
and for $1< g_{0} \le g_{1} $ fixed constants
$$
0 < g_0 \le \frac{t g'(t)}{g(t)} \le g_1,\quad\forall t > 0.
$$

If you want to know more about Orlicz Spaces any book about Orlicz Spaces has the concept or articles, for instance see the beginning of [Martinez and Wolanski - A minimum problem with free boundary in Orlicz spaces](https://arxiv.org/abs/math/0602388).


  [1]: https://i.sstatic.net/svWET.png

More specifically, let $G^p$ and $G^q$ be regular N-functions with
$$
0<g_0^p \le  \frac{t(g_p)'(t)}{g_p(t)} \le g_1^p < \infty
$$
and
$$
0<g_0^q \le  \frac{t(g_q)'(t)}{g_q(t)} \le g_1^q < \infty.
$$
Is there a relation between $g_0^p$, $g_1^p$, $g_0^q$, and $g_1^q$ which implies inclusion between $L^{G^p}$ and $L^{G^q}$?

**Obs 1** Notice that $p$ and $q$ above are indices not powers.<br>
**Obs 2** I asked this in [Mathematics StackExchange](https://math.stackexchange.com/questions/4687898/when-there-is-inclusion-of-orlicz-spaces) without answer even when I offered 250 in bounty. I believe now that here is the right place to ask. Any help is welcome, reference, direct proof etc.