# Boudedness of linear operator between generalized Orlicz spaces

I am using the notations, definitions, and results of the Section X of  on generalized Orlicz spaces.

We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is symmetric, nondecreasing, continuous, and satisfies $\varphi(0) = 0$. Then, for a measure space $(\Omega, \Sigma, \mu)$ and $\varphi$ a $\varphi$-function, $L^{\varphi}(\mu)$ is the set of measurable functions $f : \Omega \rightarrow \mathbb{R}$ such that \begin{equation} \rho_\varphi (\alpha f) := \int_{\Omega} \varphi( \alpha f ) \mathrm{d}\mu < \infty \end{equation} for some $\alpha > 0$. We call $L^\varphi(\mu)$ a generalized Orlicz space. We set \begin{align} \lVert f \rVert_\varphi = \inf \{ k > 0, \int_{\Omega} \varphi \left( \frac{\alpha f }{k}\right) \mathrm{d}\mu \leq k\}. \end{align} Then, $(L^{\varphi}(\mu),\lVert \cdot \rVert_\varphi)$ is a complete linear metric space (after identifying $f_1$ and $f_2$ when $\lVert f_1 - f_2 \rVert_\varphi = 0$), cf. , Section X, Theorem 2.

Question: Consider a continuous and linear operator $\mathrm{L}$ between two generalized Orlicz spaces $L^\varphi(\mu)$ and $L^\psi(\mu)$. When is it true that there exists a constant $C>0$ such that \begin{equation} \rho_\psi(\mathrm{L} f) \leq C \rho_\varphi(f) \end{equation} for every $f \in L^\varphi(\mu)$?

Some motivations: When the function $\varphi$ is convex, $\rho_\varphi$ defines a norm on $L^\varphi(\mu)$ that is therefore a Banach space. We talk in that case of Orlicz spaces. Here, we do not assume that $\varphi$ is convex, and not even that $\varphi$ goes to infinity at infinity. This situation occurs in the study of infinitely divisible random variables taking values in spaces of infinite dimension, see for instance . Still, $\mathrm{L}$ being continuous and linear, it is bounded for the metrics $\lVert \cdot \rVert_{\varphi}$ and $\lVert \cdot \rVert_{\psi}$ but it is not so clear to me what can we say for the quantities $\rho_{\varphi}$ and $\rho_{\psi}$ in the general case.

 M.M. Rao and Z.D. Ren (1991), Theory of Orlicz spaces

 B.S. Rajput and J. Rosinski (1989), Spectral representation of infinitely divisible processes. Probability theory and related fields, 82(3), 451-487