Given a Banach space X X , a w ∗ w^* -compact subset of X ∗ X^* , and 1 > p > ∞ 1>p>\infty , we provide an optimal relationship between the Szlenk index of K K and the Szlenk index of an associated subset of L p ( X ) ∗ L_p(X)^* . As an application, given a Banach space X X , we prove an optimal estimate of the Szlenk index of L p ( X ) L_p(X) in terms of the Szlenk index of X X . This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator A : X → Y A:X\to Y , we provide an estimate of the Szlenk index of the “pointwise A A ” operator A p : L p ( X ) → L p ( Y ) A_p:L_p(X)\to L_p(Y) in terms of the Szlenk index of A A .