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Vaughan Jones hesitated to publish his arXiv preprint since he intended to rewrite it in order to publish it as a monograph, which sadly never happened.
If embedding means "isomorphic embedding'', then a space that embeds into a reflexive space is reflexive itself. If embedding means "continuous injection'', look at the example of $\ell_1$ and $\ell_2$.
The paper R. Grząślewicz, Extreme operators on 2-dimensional $\ell_p$-spaces. Colloq. Math. 44, 309-315 (1981), Zbl 0475.47028, appears to be the first attempt at this problem. I don't know about higher dimensions, let alone $L_p(\mathbb R_+)$.
