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I am using the notations, definitions, and results of the Section X of [1] 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. [1], 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 [2]. 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.

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

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

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