I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference: https://www.researchgate.net/publication/267048828_Fractional_Integration_in_Orlicz_Spaces/link/58b43f21a6fdcc6f03fe36f7/download).

Let $\Phi$ be a Young's function and $\rho$ is a rotationally invariant norm. Sharpley characterizes a bounded operator $T_k: L_{\Phi_2} \to L_{\Phi_1}$ when there are exists $$ P: L_{\Phi_2} \to L_{\Phi_2} \quad and \quad Q:L_{\Phi_1} \to L_{\Phi_1}, $$ so the domain $L_{\Phi_2}$ is not near $L_{\infty}$ and the range $L_{\Phi_2}$ is not near $L_1$.

Find an example in which Sharpley's conditions are not satisfied (i.e the domain $L_{\Phi_2}$ is near $L_{\infty}$ and the range $L_{\Phi_2}$ is near $L_1$) and $T_k$ is still bounded.

I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference Robert Sharpley's paper, "Fractional Integration in Orlicz Spaces", Proceedings of the American Mathematical Society 59, pp. 99-106 (1976), MR0410357, Zbl 0347.46027.).

Let $\Phi$ be a Young's function and $\rho$ be a rotationally invariant norm. Sharpley characterizes a bounded operator $T_k: L_{\Phi_2} \to L_{\Phi_1}$ when there exists $$ P: L_{\Phi_2} \to L_{\Phi_2} \quad\text{ and }\quad Q:L_{\Phi_1} \to L_{\Phi_1}, $$ such that the domain $L_{\Phi_2}$ is not near $L_{\infty}$ and the range $L_{\Phi_2}$ is not near $L_1$.

I'd like to find an example in which Sharpley's conditions are not satisfied (i.e the domain $L_{\Phi_2}$ is near $L_{\infty}$ and the range $L_{\Phi_2}$ is near $L_1$) and $T_k$ is still bounded.