  We prove the operator space Grothendieck inequality for bilinear forms on
subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results
states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$
($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff
$u$ factors through a direct sum of a $p$-column space and a $p$-row space. We
also obtain several operator space versions of the classical little
Grothendieck inequality for maps defined on a subspace of a noncommutative
$L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in
[p', p]$ ($p'$ being the index conjugate to $p$). These results are the
$L_p$-space analogues of the recent works on the operator space Grothendieck
theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is
some Khintchine type inequalities for Shlyakhtenko's generalized circular
systems. One of our main tools is a Haagerup type tensor norm, which turns out
particularly fruitful when applied to subspaces of noncommutative $L_p$-spaces
($2<p<\infty$). In particular, we show that the norm dual to this tensor norm,
when restricted to subspaces of noncommutative $L_p$-spaces, is equal to the
factorization norm through a $p$-row space.
