The space L1(Lp) is primary for 1 < p < ∞

The classical Banach space L-1 (L-p) consists of measurable scalar functions f on the unit square for which parallel to integral parallel to = integral(1)(0) (integral(1)(0) vertical bar f(x, y)vertical bar(p) dy)(1/p) dx < infinity. We show that L-1(L-p) (1 < p < infinity) is primary, meaning that whenever L-1(L-p) = E circle plus F, where E and F are closed subspaces of L-1(L-p), then either E or F is isomorphic to L-1(L-p). More generally, we show that L-1 (X) is primary for a large class of rearrangement-invariant Banach function spaces.