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](https://doi.org/10.2307/2042043)", Proceedings of the American Mathematical Society 59, pp. 99-106 (1976), [MR0410357](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0410357), [Zbl 0347.46027](https://www.zbmath.org/?q=an%3A0347.46027).).

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.