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.
