Subspaces of rearrangement-invariant spaces

We prove a number of results concerning the embedding of a Banach lattice $X$ into an r.i. space $Y$. For example we show that if $Y$ is an r.i. space on $[0,\infty)$ which is $p$-convex for some $p>2$ and has nontrivial concavity then any Banach lattice $X$ which is $r$-convex for some $r>2$ and embeds into $Y$ must embed as a sublattice. Similar conclusions can be drawn under a variety of hypotheses on $Y$; if $X$ is an r.i. space on $[0,1]$ one can replace the hypotheses of $r$-convexity for some $r>2$ by $X\neq L_2.$ We also show that if $Y$ is an order-continuous Banach lattice which contains no complemented sublattice lattice-isomorphic to $\ell_2,$ $X$ is an order-continuous Banach lattice so that $\ell_2$ is not complementably lattice finitely representable in $X$ and $X$ is isomorphic to a complemented subpace of $Y$ then $X$ is isomorphic to a complemented sublattice of $Y^N$ for some integer $N.$