  Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$,
we provide an optimal relationship between the Szlenk index of $K$ and the
Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a
Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$
in terms of the Szlenk index of $X$. This extends a result of H\'ajek and
Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to
Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator
$A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.
